Chapter 1

Chapter 1. What the Socionic Compass Shows

Translator’s note on notation: the original uses Aušra’s coloured aspect glyphs (black/white symbols), which are absent from the PDF’s text layer. They are restored here from context and rendered in the standard letter notation: Ne/Se/Te/Fe for the black (extraverted) aspects of intuition, sensorics, logic and ethics, and Ni/Si/Ti/Fi for the white (introverted) ones. Where the original speaks of a symbol’s “colour,” this corresponds to the e/i subscript; its “form” corresponds to the letter (the macroaspect); the “sign” is the superscript ⁺/⁻ of the abbreviated type notation. Macroaspects are written N, S, T, F. All restored glyphs were checked against the operator algebra; none were ambiguous.

From Jung’s Compass to the Socionic Compass

The “Appendix” to C.G. Jung’s fundamental work Psychological Types [23] contains the texts of four of his papers on psychological typology, published in various years. One of them (“A Psychological Theory of Types,” 1931) was written on the basis of a lecture delivered at the Congress of Swiss Psychiatrists (Zurich, 1928). Having briefly set out the main ideas of his psychological typology, C.G. Jung concluded the text with these words:

“The overall result of my previous work in this field consists in the identification of two basic attitude types — extraversion and introversion — and four function types: the thinking, the sensing, the feeling and the intuitive, which vary according to the general attitude and thereby yield eight variants in all.

I have been asked, almost reproachfully, why I speak of exactly four functions, no more and no fewer. That there are exactly four came about, first of all, purely empirically. But that a certain degree of wholeness is achieved through them can be demonstrated by the following consideration. Sensation establishes what is actually happening. Thinking allows us to know what the given means; feeling, what its value is; and finally, intuition points to the possible ‘whence’ and ‘whither’ contained in what is present at the moment. Thanks to this, orientation in the contemporary world can be as complete as fixing one’s position in space by means of geographical coordinates. The four functions are, in a sense, the four cardinal points of the horizon — as arbitrary as they are indispensable. Nothing prevents us from shifting the point of coordinates to one side or the other, or indeed from giving them other names. Everything depends on what we agree upon, and on how expedient it is.

But I must confess one thing: on no account would I wish to do without this compass in my psychological voyages of discovery — and not for the obvious, all-too-human reason that everyone is in love with his own ideas, but because of the objective fact that it provides a system of measurement and orientation, which in turn makes possible the emergence of a c r i t i c a l p s y c h o l o g y, which we have lacked for so long” [23, pp. 643–644].

The illustration to the words of that last paragraph, shown in Fig. 1.1, was created by Andrew Joynton.

[Fig. 1.1. Jung’s Compass.]

In another of C.G. Jung’s papers from the same “Appendix” (“Psychological Typology,” 1936) we read:

“These four functions, arranged geometrically, form a cross with the axis of rationality running at right angles to the axis of irrationality.

The four orienting functions do not, of course, contain everything that is present in the conscious psyche. Will and memory, for example, are not included there. The reason is that the differentiation of these four orienting functions is, in essence, an empirical sequence of typical differences in the functional attitude” [23, p. 658].

A little further along in the text of the same paper we read:

“Thus the localization of the numinal accent gives rise to the four functional types which I first encountered in my dealings with people, but formulated systematically only much later. In practice these four types are always combined with the attitude type — that is, with extraversion and introversion — so that the functions themselves appear in an extraverted or an introverted variant. This produces a structure of eight demonstrable functional types” [23, p. 659].

It is precisely these 8 types that are examined in detail in the tenth chapter (“General Description of the Types” [23, pp. 402–495]) of C.G. Jung’s fundamental work Psychological Types. But attention should be paid to the fact that at the very end of that chapter there is a short section entitled “The Principal and Auxiliary Functions.” Here is an important quotation from it:

“For all the types met with in practice, the principle holds good that besides the conscious principal function there is a relatively unconscious auxiliary function which is in every respect different from the nature of the principal function. From these mixtures arise the familiar pictures — for instance, practical intellect paired with sensation, speculative intellect saturated with intuition, artistic intuition that selects and presents its images by means of feeling-tinged judgement, philosophical intuition which, with the help of a powerful intellect, translates its vision into the sphere of the comprehensible, and so on” [23, p. 494].

As we see, Jung allowed for a doubling of the number of types through taking the second (auxiliary) function into account. The founder of socionics, Aušra Augustinavichiute, in her work “On the Dual Nature of Man,” explaining the “paired” type names she had proposed, wrote:

“Both Jung himself and we in our own practice have become convinced that a person’s character is determined not by one function but by two. When, for example, we say ‘logical type,’ we do not know which logical one is meant: the one with sensorics, or the one with intuition. And they differ very strongly” [1, p. 10].

An example of such a “paired” name is the logical-sensory extravert (LSE). This is precisely the “logical type” that goes with sensorics — and it is a type with the extraverted attitude. In the work cited, this type was denoted symbolically as an ordered pair of symbols of the information aspects (IA) processed by the first (principal, base, program) function and the second (auxiliary, creative) function: (Te, Si). The block formed by the first and second functions received the name Ego block. By the rules of “blocking,” once the IA processed by the first function is fixed, only two variants remain for the IA of the second. Explanations of these rules will be given later.

The socionic type just mentioned — the type of information metabolism (TIM) — can also be denoted as Te⁺. The simple rule for passing to such abbreviated symbolic notations of TIMs will be described below. Here let us only note that, at Aušra Augustinavichiute’s suggestion, small changes were made to the names of the functions. Instead of the four {intuition, sensation, thinking, feeling}, socionics came to use the four {intuition, sensorics, logic, ethics}. The corresponding so-called macroaspects of the information flow bear the same names: N (intuition), S (sensorics), T (logic), F (ethics). Each macroaspect divides into two IA, whose symbolic notations differ in colour (black and white).

There are three more TIMs with logic in first place: LIE (Te, Ni — abbreviated Te⁻), LSI (Ti, Se — abbreviated Ti⁺), LII (Ti, Ne — abbreviated Ti⁻). The whole quartet of these types can be denoted T. Introducing analogous abbreviated symbolic notations for the remaining TIMs and their corresponding quartets, it is easy to obtain the “folded” variant of the Socionic Compass I propose (Fig. 1.2), which will be reminiscent of Jung’s Compass. Just as in Jung’s Compass, the Socionic Compass has an axis of irrationality (N — S) and an axis of rationality (T — F). From the example given of the quartet of first-function logicians it should be clear what the Ego blocks of all 16 TIMs making up the socion may look like.

[Fig. 1.2. The “folded” variant of the Socionic Compass.]

The main aim of the present chapter is to show how to “unfold” the “folded” variant of the Socionic Compass presented in Fig. 1.2, extracting from it information about the composition and structure of three 16-element groups (in the mathematical sense of the word), to the study of whose properties a considerable number of socionic publications have been devoted. These are the group of operators of the classical intertype relations (IR) and two groups of central bipolar traits of TIMs. These two groups of traits are singled out — precisely from the standpoint of the classical IR — from the vast set of groups of central sections of the socion having the same order.

A System of Formalized Notation for the Operators of the Classical Intertype Relations

In this section it will be shown how to obtain the group of operators of the classical IR by means of the Socionic Compass. By the classical model of the IR we shall understand the one proposed by Aušra Augustinavichiute in her work “The Theory of Intertype Relations” [1, pp. 109–181]. That model is not directly tied to any group of bipolar traits of TIMs.

The classical IR are often examined by invoking the Models “A” of the communicants, although in the work just cited the founder of socionics used the simplified Models “Yu” for the TIMs. We, for our part, shall now use the abbreviated symbolic notations of the TIMs as their models. With this approach one obtains exactly the same group of IR operators.

Let us note that it was G.R. Reinin who proposed bringing in the mathematical concept of operator in the treatment of intertype relations [19, p. 174]. By means of an IR operator, the model of one TIM is transformed into the model of another. If, in a pair of TIMs, one and the same operator transforms the model of the first communicant into the model of the second and the model of the second into the model of the first, such relations are called symmetric in socionics. If the operators do not coincide, the IR are said to be asymmetric. In other words, in the case of symmetric IR a single operator “works” in the pair of TIMs — an operator that is its own inverse — while in the asymmetric case there are two different operators that are inverses of each other.

If a pair of communicants consists of two identical TIMs, one speaks of identity relations, and the corresponding operator is called the identity operator. We shall denote this operator by the symbol “1”. In the group of IR operators it plays the role of the identity element. In the socionics literature this operator may be denoted “T”, “TO”, “TZh”. Obviously, notations of that sort are tied to the words used in naming the IR. But they do not reflect the mathematical structure of the group of operators, and they are not adapted to carrying out the binary operation in that group. From their outward appearance one cannot tell how the models of the socionic types transform under their action. In our formalized system of notation, the visual images of the operators carry far more information about their mathematical properties.

The set of generators of the group of operators of the classical IR can be reduced to three elements. This is clearly visible in Fig. 1.3.

[Fig. 1.3. The triple of operators {1*, c, m} of symmetric IR generates a 16-element non-commutative group.]

In this variant of the Socionic Compass there are connecting lines of only three kinds (1*, c, m). Nevertheless, along these connecting lines one can get from the symbol of any TIM to any other. This means that the triple of operators {1*, c, m} can serve as a generating system of the 16-element non-commutative group of operators of the classical IR. Admittedly, the expressions for the other operators in terms of the operators of this triple may turn out to be quite long.

We shall compare our formalized notations for the operators of the classical IR with those used in V.V. Gulenko’s book Humanitarian Socionics [6], which adopts a system in which the notations of at least four operators of asymmetric IR are visually distinguished.

The notation 1* indicates that only the sign used as the superscript is to be changed. For example, 1*(Fe⁺) = Fe⁻. With unformalized notations of the IR operators and three-letter TIM codes this formula would look like this: KIN(EIE) = ESE, and the corresponding sentence would be: the kindred type of the ethical-intuitive extravert is the ethical-sensory extravert. But one can also say: EIE and ESE are kindred. Such a substitution is possible because the operator 1* is its own inverse (it is an involution). In other words, the kindred IR are symmetric. From the standpoint of how the abbreviated symbolic notation of a TIM changes under the action of the operator 1*, this fact is perfectly obvious: changing the sign twice, we return to the original TIM. The following sentence is also correct in socionics: the kindred of one’s kindred is one’s identical. Mathematically one can write: (1*)⁻¹ = 1*, or 1* · 1* = (1*)² = 1. It is precisely the taking into account of the second function that makes it possible to distinguish kindred types — which leads from Jung’s 8-element typology to the 16-element socionic typology.

Another operator for which a formalized notation is introduced in Fig. 1.3 is the operator of extinguishment (complete opposition, neutralization). The action of this operator on the abbreviated symbolic notation of a TIM is easy to see. It changes only the colour of the symbol, leaving unchanged not only the form but also the superscript (the sign). For this reason the letter “c” (from the English word colour) has been used to denote this operator. If this letter is read not as “cee” but as “tse,” it may remind the Russian-speaking reader of the word tsvet (colour). This operator too is its own inverse (an involution). Indeed, changing the colour twice, we return to the symbol of the original TIM.

The last element of the generating set under consideration for the 16-element group of operators of the classical IR is the mirror operator (m, from the English mirror), which changes the sign, the colour and the form — and changes the form depending on the sign of the original TIM. There will be no problems with this operator once the skill of handling the abbreviated symbolic notations of the TIMs has been worked up. One must learn to expand them mentally into the unfolded form (as ordered pairs of the IA symbols of the first and second functions) and to perform the inverse operation of “folding.” Mirror TIMs are those whose Ego blocks, consisting of the first two functions, differ in the order of the IA symbols. For example, m((Ne, Ti)) = (Ti, Ne). After replacing these notations of the TIMs by the abbreviated ones, the same equality is rewritten as: m(Ne⁺) = Ti⁻.

Before proceeding to the remaining elements of the group of operators of the classical IR, let us give the promised explanations concerning the rule for the use of signs in the abbreviated symbolic notations of TIMs. Since these notations are not used very often in the socionics literature, we shall show, with the help of the Socionic Compass, how the sign in a TIM’s abbreviated notation lets one find the IA of its second function.

Look once more at Fig. 1.3, where the Socionic Compass is presented so that the abbreviated symbolic notations of all 16 TIMs making up the socion are visible. If the superscript in a TIM’s abbreviated symbolic notation is the sign ”+”, then the information aspect of its second function will be the same as that of the first function of the nearest TIM when moving clockwise along the arc of the corresponding circle. In the case of the sign ”−”, one must move counterclockwise. Note that in both cases the movement proceeds along the arc connecting the symbols of two mirror partners (m).

Take, for example, Fe⁺, whose abbreviated symbolic notation carries the sign ”+“. Moving clockwise along the corresponding circle, we run into Ni⁻. This means that Fe⁺ = (Fe, Ni) = EIE. Now consider Ti⁻ as an example of a TIM with the other sign. Moving counterclockwise along the corresponding circle, we run into Ne⁺. Conclusion: Ti⁻ = (Ti, Ne) = LII.

Running ahead, let us point out that the sign in the abbreviated symbolic notation records the TIM’s membership in a pole of the bipolar trait right / left (+ / − respectively). This indication may prove useful to those for whom a clockwise turn (of a car’s steering wheel, say) is associated with turning right, and a counterclockwise turn with turning left. Note that the linkage between the forms of the IA symbols of the first and second functions does not depend on “colour”: if Fe⁺ = (Fe, Ni), then Fi⁺ = (Fi, Ne) as well.

One more anticipatory remark may help with the correct use of signs in the abbreviated symbolic notations of TIMs. Important in socionics is the division of the socion into the following quartets of TIMs:

• {Ne⁺, Ti⁻; Ni⁺, Te⁻} = {ILE, LII; ILI, LIE} — the Club of Scientists;
• {Ne⁻, Fe⁺; Ni⁻, Fi⁺} = {IEE, EIE; IEI, EII} — the Club of Humanitarians;
• {Se⁺, Fe⁻; Si⁺, Fi⁻} = {SEE, ESE; SEI, ESI} — the Club of Socials;
• {Se⁻, Te⁺; Si⁻, Ti⁺} = {SLE, LSE; SLI, LSI} — the Club of Managers.

It is easy to notice that each “professional club” can be split into two mirror pairs and two extinguishment pairs. If one remembers, for instance, that among the Scientists intuition goes with “plus” and logic with “minus,” then placing the remaining signs should cause no difficulty. The names of the quartets suggest that in the Ego blocks of the Scientists intuition is united with logic. Among the Humanitarians, intuition is united with ethics, so their intuition must carry the other sign, i.e. “minus.” Accordingly, ethics among the Humanitarians will carry “plus,” since the signs of the IA within a block are different. Ethics is a “strong” function among the Socials too, but with them it is united with sensorics, which is marked by a “minus” — while sensorics itself among the Socials carries a “plus.” Correspondingly, the Managers will have sensorics with a minus, and logic (unlike the Scientists) with a “plus.”

Let us note in passing that the TIMs of the Scientists’ and Socials’ clubs are called democrats in socionics, and the TIMs of the Managers’ and Humanitarians’ clubs aristocrats. Relying on this, one can observe that among the democrats the irrational functions carry “plus” and the rational ones “minus,” while among the aristocrats it is the rational functions that carry “plus.”

Later we shall see that the dichotomy of the socion by the bipolar trait democrats / aristocrats is just as important from the standpoint of the classical IR as the dichotomy by the trait irrationals / rationals. Note that for individual information aspects (disregarding sign) there is no division into democratic and aristocratic: such a division appears only at the level of the block consisting of two IA.

At the same time, the division into irrational and rational exists already at the level of the macroaspects and carries over to the individual IA. By the rules of “blocking,” a block contains IA of different -nality, as one says — i.e. one IA is irrational and the other rational. Nevertheless, the bipolar trait irrational / rational can be extended to blocks as well, if they are treated as ordered pairs of IA (or, correspondingly, pairs of functions). The pole of this trait for a block is determined by the pole of the first element of the block.

Consider the mirror pair of TIMs {Ne⁺, Ti⁻}. The Ego block of each of these types contains the same IA (Ne and Ti), but the order of these elements in the block differs. Going by the first element, the ILE (Ne, Ti) is considered an irrational, and the LII (Ti, Ne) a rational. The situation is analogous with the bipolar trait extraverts / introverts, whose pole is fixed by the colour of the IA symbol of precisely the first function: Ne⁺ is an extravert, Ni⁺ an introvert.

Let us return to the operators of the classical IR. Given two operators, one can “multiply” them, introducing a binary operation on the set of operators. The attentive reader may have noticed that we have already, without special warning, made use of the binary operation in multiplying the operator 1* by itself. But when we need to multiply two different operators, the order of the factors may turn out to matter. The question arises: are the kindred of one’s extinguisher (1* · c) and the extinguisher of one’s kindred (c · 1*) one and the same thing?

That one should be careful with questions of this kind is suggested by our experience of ordinary spoken language. It is clear, after all, that “the son of one’s sister” is not the same as “the sister of one’s son.” But in our particular case the product of the operators 1* and c does not depend on the order of the factors. In both cases we obtain the operator of the mirage relations, for which we introduce the notation c*. In such cases one says that the operators commute with each other. Hence both the equality c* = 1* · c and the equality c* = c · 1* are valid.

The commutativity of the operators 1* and c just noted is completely obvious from the standpoint of our formalized notations. Indeed, everyone understands that the operations of changing the sign and changing the colour can be interchanged: the result will not change.

Note that within each quartet of TIMs there are 6 pairs. Take, for example, the quartet of first-function intuitives (N). In Fig. 1.3, the operators of the relations are shown by the corresponding connecting lines in only three of the pairs. But this is quite sufficient to understand what the relations are in the remaining three. Acting with the operators {1*, c, c*} on any TIM of the quartet in question, we obtain the other three. For example, 1*(Ne⁺) = Ne⁻, c(Ne⁺) = Ni⁺, c*(Ne⁺) = Ni⁻. Let us underline the fact that these operators, like 1, do not change the form of the IA symbol of the first function of the TIM on which they act.

Let us pass on to Fig. 1.4. It is intended to introduce the formalized notations of the remaining operators. All of them, like the mirror operator (m), change the form of the first-function IA symbol of the TIM on which they act. We begin with the operators of superego (-1) and conflict (-m), for which special kinds of connecting lines are provided in the proposed variant of the Socionic Compass (see Fig. 1.4).

[Fig. 1.4. The operators of the classical intertype relations on the “unfolded” variant of the Socionic Compass.]

The superego operator (-1) changes neither the sign nor the colour, and changes the form in an entirely unambiguous way (within a single -nality, as one says). One may say that when this operator is used, the initial and resulting TIMs belong to one and the same pole of the dichotomy irrationals / rationals. The notations of TIMs linked by the superego relation (-1) are located on the diagram symmetrically with respect to its centre.

As one can already guess, the conflict operator (-m) is the product of the superego (-1) and mirror (m) operators, the order of the factors being immaterial. The superego of one’s mirror partner is one’s conflictor, and the mirror partner of one’s superego is likewise one’s conflictor. The commutativity of the superego (-1) and mirror (m) operators is easily explained in the language of transformations of the TIM notations. One need only note that under the action of the superego operator (-1) the form of the IA symbol changes, within a single -nality, not only for the first function but for the second as well. For example, -1((Ne, Ti)) = (Se, Fi). It now becomes perfectly obvious that the operation of changing the form within a single -nality (-1) and the operation of interchanging the IA symbols of the first and second functions (m) can be interchanged.

By means of the operators {-1, m, -m}, from one TIM one can recover the other three elements of the quartet whose symbols lie on one and the same circle of the Socionic Compass. For example, -1(Ne⁺) = Se⁺, m(Ne⁺) = Ti⁻, -m(Ne⁺) = Fi⁻. Recalling which quartets of TIMs are linked by the operators of the triple {1*, c, c*}, it is easy to see that in the Socionic Compass one can get from the symbol of any TIM to the symbol of any other in at most two “steps,” using the operators of the six already examined: {1*, c, c*, -1, m, -m}.

When two “steps” are needed, let the first be accompanied by a change of the symbol’s form (corresponding to the operators of the triple {-1, m, -m}), while on the second the form does not change (corresponding to the triple {1*, c, c*}). Geometrically this can be pictured as movement along a circle on the first “step” and along a radius on the second. Of course, if the superego operator (-1) has to be used on the first “step,” the movement along the circle can be replaced by a “leap” along the diameter, from the initial point to the point symmetric to it with respect to the centre of the Socionic Compass.

In the upper part of Fig. 1.4 are written out the expressions for five more operators of the classical IR: business (-1*), dual (-c), semi-dual (-c*), quasi-identity (cm) and activation (-cm). They are presented as products of two factors, one taken from the triple {-1, m, -m} and the other from {1*, c, c*}. So far only cases have been considered in which the operators being multiplied commute with each other — in other words, the product does not depend on the order of the factors. Moreover, each of the resulting operators is its own inverse, i.e. it is an operator of symmetric IR. These assertions are fairly obvious once one takes into account the transformations of the abbreviated symbolic notations of the TIMs produced by the operators entering the products.

Let us now consider the operators that can be introduced by the formulas presented in the lower part of Fig. 1.4. Note the operator m* (of passage to one’s supervisee). This operator is obtained as the product of the non-commuting operators 1* and m, taken in a definite order. The supervisee is the kindred of one’s mirror partner: m* = 1* · m. For example, m*(Ne⁺) = 1*[m(Ne⁺)] = 1*[Ti⁻] = Ti⁺. Let us find, by the same method, the supervisee of Ti⁺: m*(Ti⁺) = 1*[m(Ti⁺)] = 1*[Se⁻] = Se⁺. As we see, repeated application of this operator has not returned us to Ne⁺. Bearing in mind that Se⁺ = -1(Ne⁺), we can write the equality: (m*)² = -1. On being squared (multiplied by itself), this operator has yielded not the identity operator (1) but the superego operator (-1).

Let us try changing the order of the operators in the product — that is, finding the mirror partner of the kindred of Ne⁺. Here is what results: m[1*(Ne⁺)] = m[Ne⁻] = Fi⁺. The notation for the corresponding operator (-m*) is introduced in view of the fact that Fi⁺ = -1(Ti⁺). Regardless of the initial TIM, the following equality holds:

m · 1 = -1 · m,**

which has been called the fundamental commutation relation (the interchange rule) [13]. The resulting operator (-m*) transforms Ne⁺ into Fi⁺. On the other hand, it is easy to check that the equality m*(Fi⁺) = Ne⁺ holds.

Thus the operator -m* is the inverse of the operator m*, which is written mathematically as:

(m*)⁻¹ = -m*, or -m* · m* = m* · (-m*) = 1.

In other words, -m* is the operator of passage to one’s supervisor (controller). The square of this operator, like that of m*, equals the superego operator (-1).

One more pair of mutually inverse operators of asymmetric IR can be obtained by multiplying the operators of the pair just examined by the extinguishment operator (c):

• cm* = c · m* (the operator of passage to one’s benefactor);
• -cm* = c · (-m*) (the operator of passage to one’s beneficiary).

One need only note that the benefactor is not the extinguisher of one’s supervisor but the extinguisher of one’s supervisee (c · m*) — or, equivalently, the dual of one’s supervisor: cm* = -c · (-m*).

For the operators of passage to the supervisor (-m*), the benefactor (cm*) and the beneficiary (-cm*), other expressions are given in Fig. 1.4. This is because we wished, as in the previous cases, to show the possibility of making the required passages in two “steps,” the first of which corresponds to an operator from the triple {-1, m, -m} (movement in the Socionic Compass along a circle), while on the second “step” an operator from the triple {1*, c, c*} is used (movement along a radius). On the first “step” the form of the symbol changes; on the second it does not.

At the beginning of this section, in connection with the discussion of Fig. 1.3, it was said that the triple of operators {1*, c, m} can serve as a generating system of the 16-element non-commutative group of operators of the classical IR. But if the expressions for the other elements of the group are written directly in terms of the operators of this triple, they may turn out to be quite long.

Suppose we need to “get” from Ne⁺ to Si⁺ using only the operators of the triple {1*, c, m}. Here is one possible — and not the longest — path: Ne⁺ → Ne⁻ → Ni⁻ → Fe⁺ → Fe⁻ → Si⁺. The sequence of operators effecting the required transformations gives the following expression for the duality operator (-c), which transforms Ne⁺ into Si⁺ directly: m · 1* · m · c · 1*. One must only remember that with this way of writing a product of operators it is understood that the transformations are carried out in reverse order (from right to left) — that is, first 1*(Ne⁺) = Ne⁻, then c(Ne⁻) = Ni⁻, m(Ni⁻) = Fe⁺, and so on.

Let us demonstrate, on this same example, that the use of the formalized notations of the IR operators makes it possible to compute such products in one’s head. The operator c commutes with all operators; consequently it can be moved to any position. All the operators in our generating system are involutions (their squares equal 1). Multiplying any operator by the identity operator (1) leaves it unchanged. Finally, one must not forget the fundamental commutation relation (m · 1* = -1* · m). Taking these simple rules into account, we obtain:

m · 1* · m · c · 1* = −(1* · (m · m) · 1*) · c = -c.

To be honest, we also made use of the fact that the superego operator (-1), represented by the “minus,” can likewise be moved to any position — like c, it commutes with all operators. In addition, we used the associativity property, obligatory for groups, which allowed us to place brackets where needed, indicating which “stretches” of the expression can be “deleted” on the grounds that the squares of the operators m and 1* equal the operator 1.

If one uses the extended generating system whose fourth element is the superego operator (-1), then for any operator of the classical IR one can write its canonical representation in the form:

(-1)ᵏ · cˡ · (1)ᵖ · mᑫ,*

where the exponents take only the values 0 or 1 [13]. In other words, in such a representation of any operator, each element of the quadruple {-1, c, 1*, m} occurs at most once. If all the exponents are zero, we have the identity operator (1). The longest path, if one uses only the operators of this quadruple, consists of four “steps” and corresponds to the operator of passage to the beneficiary (-cm*). In that case the operator m acts on the first “step”; the order of action of the remaining operators after that no longer matters, since they commute among themselves.

In discussing Fig. 1.4 it was said that in the Socionic Compass one can get from the symbol of any TIM to the symbol of any other in at most two “steps,” using on the first “step,” for movement along a circle, operators from the triple {-1, m, -m}, and on the second “step,” for movement along a radius, operators from the triple {1*, c, c*}. For the operator of passage to the beneficiary (-cm*), the first “step” corresponds to the conflict operator (-m), and on the second “step” one must perform the transformation described by the operator c* (of the mirage IR). Indeed, c* · (-m) = -cm*.

In effect, the point is to decompose the canonical representation of any operator into two factors, each of which is enclosed in square brackets in the following expression:

(-1)ᵏ · cˡ · (1*)ᵖ · mᑫ = [(1*)ᵖ · cˡ] · [(-1)ᵏ · mᑫ].

Such a decomposition into two factors makes it easier to find, in one’s head, the operator that transforms the initial TIM into the resulting one. The factor [(-1)ᵏ · mᑫ] secures the required form of the resulting TIM’s symbol, while the factor [(1*)ᵖ · cˡ], corresponding to the second “step,” produces the required colour and sign.

Consider a concrete example of the problem of finding the IR operator from a known initial and resulting TIM. What operator must act on Ne⁺ to yield Te⁻? To obtain the required form, of the operators of the form [(-1)ᵏ · mᑫ] only m fits (k = 0, q = 1): m(Ne⁺) = Ti⁻. It remains only to change the colour. This is the “responsibility” of one of the operators of the form [(1*)ᵖ · cˡ] — namely c (p = 0, l = 1). The sought operator is thus found as the product c · m = cm; that is, we are dealing with the quasi-identity operator.

And here is an example of finding the resulting TIM from the initial TIM and the operator. How does one find the benefactor of the LII? Since the operator cm* is representable as the product c* · m, two “steps” are needed: c*[m(Ti⁻)] = c*[Ne⁺] = Ni⁻; that is, the benefactor of the LII is the IEI.

Finally, we must work through an example of finding the initial TIM from the resulting TIM and the operator. We shall show that such a problem is easily reduced to the previous kind. If the operator is an involution (i.e. the IR are symmetric), one simply acts with the given operator on the resulting TIM. Compare two questions:

1. For which TIM is the IEE the semi-dual?
2. Which TIM is the semi-dual of the IEE?

In both cases the answer is the same, since the semi-dual relations are symmetric. Therefore, in answering the first question we may proceed exactly as in answering the second, using the representation of the given operator in the form -c* = c* · (-1). We then obtain:

-c*(Ne⁻) = c*[-1(Ne⁻)] = c*[Se⁻] = Si⁺.

One may say that the IEE and the SEI are semi-duals.

The case of asymmetric IR is not much harder. Suppose the question is: for which TIM is the IEE the supervisor? One simply reformulates it thus: which TIM is the supervisee of the IEE? And then one proceeds by the old algorithm:

m*(Ne⁻) = 1*[m(Ne⁻)] = 1*[Fi⁺] = Fi⁻.

The answer to the original question is the same TIM as for the reformulated one. The IEE is the supervisor of the ESI. Put differently, the ESI is the supervisee of the IEE.

Partitioning the Socion by Means of the Proper Subgroups of the Group of Operators of the Classical IR

The formalized notations of the operators of the classical intertype relations, like the symbolic notations of the TIMs and the Socionic Compass itself, are useful elements of infographics, allowing one not only to present socionic information in a compact and visual form, but also to derive consequences in a simpler and more transparent way. How can one quickly tell whether two given operators of the classical IR commute with each other? It turns out this can easily be done from the outward appearance of their formalized notations.

Looking attentively at the proposed notations of the IR operators, it is not hard to notice four bipolar traits connected with the presence / absence of the following elements: 1) the letter m; 2) the “asterisk” (*); 3) the letter c; 4) the “minus” (-). Each of these traits divides the whole set of formalized notations exactly in half, and the corresponding dichotomies are independent and pairwise orthogonal. Specifying a concrete set of poles of these bipolar traits uniquely determines an operator of the classical IR.

Note that the formalized notation of every operator of asymmetric IR necessarily contains both the letter m and the “asterisk” () — and this singles those operators out against the background of the rest. On the other hand, there are four operators whose notations contain neither the letter m nor the “asterisk” (). The quadruple {1, -1, c, -c} gathers together precisely those operators that commute not only among themselves but with all the others. This quadruple is itself a group in the mathematical sense of the word and, accordingly, a subgroup of the whole 16-element non-commutative group of operators of the classical IR. The subgroup gathering all the elements that commute with the rest is called the centre of the group. The whole group can be decomposed into cosets with respect to the centre. In the present case the decomposition is:

1. {1, -1, c, -c}; 2) {1*, -1*, c*, -c*};
2. {m, -m, cm, -cm}; 4) {m*, -m*, cm*, -cm*}.

As we see, under this decomposition all the operators of asymmetric IR automatically gather in the last coset. In the coset {1*, -1*, c*, -c*} the notations of all the operators contain the “asterisk” (*) but not the letter m; in the coset {m, -m, cm, -cm} the situation is the reverse. Note also that in every coset two of the four operators contain the letter c in their notations, and two the “minus” (-). This is exactly where the independence and orthogonality of the central sections of the set of operator notations by the bipolar traits indicated above is reflected.

If one takes any element of the last three cosets and multiplies it by the elements of the centre, one obtains the whole of the corresponding coset — this, in fact, is how the decomposition of the whole group into cosets with respect to its centre is carried out. Operators not belonging to the centre commute only with the operators of the centre and of their own coset. When two operators from one and the same coset are multiplied, the product belongs to the centre. If one multiplies two operators from different non-central cosets, the product belongs to the third non-central coset. Since an operator’s membership in a particular coset with respect to the centre is plainly visible from the outward appearance of its formalized notation, the question of whether two particular operators commute or not should present no difficulty.

Let us point out one more circumstance worth keeping in mind. If two operators of the classical IR do not commute with each other, then their two possible products differ only in sign (interchanging the factors is equivalent to multiplication by the superego operator (-1)). In other words, for any two non-commuting operators there holds an interchange rule analogous to the fundamental commutation relation. Such interchanges can be useful in finding the result of the binary operation. For example:

m · m* = -m* · m = -1*; -cm · 1* = 1* · cm = cm*; etc.

Using the formalized notations of the operators of the classical IR, it is not difficult to find all the proper subgroups of their 16-element non-commutative group. We shall be concerned with subgroups of orders 8, 4 and 2. By means of these subgroups the socion can be divided into the corresponding number of parts (2, 4 and 8). Under such divisions, one and the same set of IR operators is realized in every part, allowing one, from a single TIM of a given part, to find all the others belonging to that part of the socion. The set of operators is obtained from the corresponding subgroup by excluding the identity element, i.e. the identity operator (1).

The most obvious case is that of the subgroups of order 2. Each such subgroup contains the identity operator (1) and one further operator of symmetric IR. And of the latter there remain 11:

-1, c, -c, 1*, -1*, c*, -c*, m, -m, cm, -cm.

Correspondingly, there exist 11 octochotomies of the socion into pairs of TIMs, joined into those pairs by one of the operators listed. In other words, in this way the socion can be divided either into 8 superego pairs (-1), or into 8 extinguishment pairs (c), or into 8 dual pairs (-c), or into 8 kindred pairs (1*), and so on.

Of subgroups of order 8 (and the corresponding dichotomies of the socion) there are 7:

1. {1, -1, c, -c, 1*, -1*, c*, -c*},
2. {1, -1, c, -c, m, -m, cm, -cm},
3. {1, -1, c, -c, m*, -m*, cm*, -cm*},
4. {1, -1, 1*, -1*, m, -m, m*, -m*},
5. {1, -1, 1*, -1*, cm, -cm, cm*, -cm*},
6. {1, -1, c*, -c*, m, -m, cm*, -cm*},
7. {1, -1, c*, -c*, cm, -cm, m*, -m*}.

The first three subgroups of this list contain the centre of the whole group together with one entire coset with respect to the centre. The third differs from the first two in that all 4 operators of asymmetric IR have entered it, whereas the first two consist exclusively of operators of symmetric IR. Each of the subgroups numbered 4–7 in the list contains two elements from every coset with respect to the centre, differing from each other in sign. It is obvious that the group of operators of the classical IR has no other 8-element subgroups. And this means that there exist exactly 7 central dichotomies of the socion such that each half of the socion can be reconstructed from any single representative of it by means of the operators of the corresponding subgroup of order 8.

For now let us consider just one illustrative example. Take Ne⁺ and act on it with the operators of the first subgroup of our list. We obtain the following sequence of eight TIMs: {Ne⁺, Se⁺, Ni⁺, Si⁺, Ne⁻, Se⁻, Ni⁻, Si⁻}. If we perform the same procedure taking as the initial TIM, say, Se⁻, we obtain the sequence {Se⁻, Ne⁻, Si⁻, Ni⁻, Se⁺, Ne⁺, Si⁺, Ni⁺}. As expected, the composition of the eight TIMs has not changed. In both cases what has been written out are all the irrationals. Had the initial TIM been a rational one, the result would have been the eight rationals. Acting with the same sequence of operators on, say, Ti⁻, we obtain the following sequence of TIMs: {Ti⁻, Fi⁻, Te⁻, Fe⁻, Ti⁺, Fi⁺, Te⁺, Fe⁺}. Thus the first order-8 subgroup of our list can be put into one-to-one correspondence with the bipolar trait irrationals / rationals, which verbalizes the corresponding central dichotomy of the socion. We shall return to the verbalization of the remaining six dichotomies.

For the moment, let us make the differences among the seven order-8 subgroups under discussion visually apparent. To this end we make use of the so-called Cayley graphs [5], which for our case are presented in Fig. 1.5. The first two 8-element subgroups of our list can be represented as the direct product of three of their cyclic subgroups of order 2.

In the first case one such triple is {1, -1}, {1, c}, {1, 1*}; in the second, {1, -1}, {1, c}, {1, m}. Accordingly, the minimal generating system consists in both cases of three elements ({-1, c, 1*} and {-1, c, m} respectively). All 8 elements of the group are represented by the vertices of the graph (see the left-hand graph in Fig. 1.5). The operators belonging to the generating system are additionally represented by the edges of the Cayley graph, distinguished from one another in some way (for instance, by colour). At every vertex, edges of all three kinds meet. If some vertex of the Cayley graph is assigned the identity element of the group (1), and each kind of edge its own element of the generating system, then the correspondence between the remaining vertices of the graph and the elements of the group is thereby fixed as well.

[Fig. 1.5. Cayley graphs for the three kinds of 8-element subgroups of the group of operators of the classical IR.]

The third subgroup of order 8 is unique of its kind; the middle graph of Fig. 1.5 corresponds to it. It is representable as the direct product of two of its cyclic subgroups, one of order 4 and the other of order 2 — for example, {1, -1, m*, -m*} and {1, c}. The subgroup of type C4 can in this case be generated by the single operator m*. The generating system of the whole 8-element group can therefore be reduced to two elements ({m*, c}). Since the operator m* is not an involution (not its own inverse), it is represented on the Cayley graph by directed edges (with an arrow).

The remaining four subgroups, numbered 4–7 in the list, are 8-element dihedral groups (D₈).¹ The right-hand graph of Fig. 1.5 is the simplest variant of the Cayley graph for such groups. The generating system can in this case be reduced to two involutions that do not commute with each other. Let us assign to the last four numbers of our list of 8-element subgroups possible two-element generating systems:

4 — {1*, m}; 5 — {1*, cm}; 6 — {c*, m}; 7 — {c*, cm}.

Let us now pass to the 4-element subgroups. There are 15 subgroups of this order in all. Only two of them are cyclic subgroups of the fourth order (C4):

{1, -1, m*, -m*} and {1, -1, cm*, -cm*}.

For these groups the minimal generating system consists of just one element, which is not an involution.

The remaining 13 subgroups of order 4 contain no operators of asymmetric IR and can be represented as the direct product C2×C2. Fig. 1.6 shows the corresponding Cayley graphs for the two kinds of subgroup indicated.

[Fig. 1.6. Cayley graphs for the two kinds of 4-element subgroups of the group of operators of the classical IR.]

Although the 13 subgroups of order 4 are isomorphic to one another, it is useful to distinguish them by which operators of the triple {-1, c, -c} enter their composition. All three of these operators together enter only the centre of the group ({1, -1, c, -c}).

Exactly one operator of this triple enters the following subgroups of order 4:

1. {1, -1, 1*, -1*}, {1, -1, c*, -c*}, {1, -1, m, -m}, {1, -1, cm, -cm} — containing the superego operator (-1);
2. {1, c, 1*, c*}, {1, c, -1*, -c*}, {1, c, m, cm}, {1, c, -m, -cm} — containing the extinguishment operator (c);
3. {1, -c, 1*, -c*}, {1, -c, -1*, c*}, {1, -c, m, -cm}, {1, -c, -m, cm} — containing the duality operator (-c).

Let us demonstrate how a tetrachotomy of the socion is effected by means of the fourth-order subgroups of the group of operators of the classical IR, taking as our example the division into quadras — i.e. into those quartets of TIMs in which the relations of duality (-c), mirror (m) and activation (-cm) are realized. Clearly the subgroup in question is {1, -c, m, -cm}. Taking an arbitrary TIM and acting on it with the operators of this subgroup, we obtain one of the quadras. Next one takes a TIM not obtained in this way and acts on it with the operators of the same quadruple. After the third cycle, the TIMs of the last quadra remain. Of course, with such a procedure the order in which the quadras appear will be random, but the division of the socion will in any case be into quadras and not into any other quartets of TIMs. The quadras are denoted by Greek letters in the following order:

1. {Ne⁺, Si⁺, Fe⁻, Ti⁻} — α; 2) {Fe⁺, Ti⁺, Se⁻, Ni⁻} — β;
2. {Se⁺, Ni⁺, Te⁻, Fi⁻} — γ; 4) {Te⁺, Fi⁺, Ne⁻, Si⁻} — δ.

In an analogous way the socion can be divided into 4 parts by means of the remaining 14 subgroups of order 4. The quartets of TIMs obtained by such divisions G.R. Reinin proposed to call homogeneous [19, p. 172] — meaning that in such a quartet all the constituent TIMs find themselves “in an identical intertype situation.” Note that among the homogeneous quartets (from the standpoint of the classical IR) fall also those 8 quartets corresponding to the subgroups of type C4, which consist of two mutually inverse operators of asymmetric IR (m* and -m*, or cm* and -cm*), the superego operator (-1), and the identity operator (1) as the identity element of the group. The identity operator (1) is, of course, not realized in quartets consisting of distinct TIMs; it was therefore not included in the “formula” of a quartet [16]. Since each of the 15 “formulas” obtained by discarding the identity element from the subgroups of order 4 corresponds to 4 quartets of TIMs, there exist 60 homogeneous quartets in all.

The division of quartets of TIMs into homogeneous and non-homogeneous arose because G.R. Reinin divided the socion into 4 parts in a fundamentally different way — by using pairs of bipolar traits from the group he had constructed, whose generating system was the quadruple extraverts / introverts, irrationals / rationals, intuitives / sensorics, logicians / ethicists. This quadruple is usually called the “Jung basis,” although its last two elements could not, in principle, have appeared in Jung’s 8-element typology, in which kindred types were not distinguished [15].

When the socion is divided by means of pairs of ARTs (Augustinavichiute–Reinin traits), the quartets of TIMs turn out homogeneous in only 15 of the 35 possible cases. And of those 15, only in 11 cases was there a single “formula” common to all four quartets. Four tetrachotomies of the socion organized by means of pairs of ARTs proved to be “two-formula” ones: for quartets consisting of irrationals, the “formula” differed from the one corresponding to the quartets of rationals. By means of the ARTs one cannot organize the “single-formula” tetrachotomies of the socion corresponding to the following subgroups of order 4:

{1, c, 1*, c*}, {1, c, -1*, -c*}, {1, -c, 1*, -c*}, {1, -c, -1*, c*}.

Clearly, any tetrachotomy of the socion into four quartets of TIMs can be obtained by means of two suitable central dichotomies. This means that besides the group of ARTs there must exist at least one more group of bipolar traits that is of interest from the standpoint of the classical intertype relations. Such a group was constructed and received the name of the group of JMTs (Jung–Minaev traits) [11]. Its construction was connected with the wish to verbalize precisely those dichotomies of the socion whose pairs make it possible to organize the tetrachotomies of the socion corresponding to the 4-element subgroups of the 16-element non-commutative group of operators of the classical IR just listed.

In the next section it will be shown how both groups of bipolar traits (the ARTs and the JMTs) can be obtained directly from the Socionic Compass, which was built on the basis of Jung’s Compass. We shall begin with the group of JMTs, in which — unlike the far better-known group of ARTs — one can find a triple of bipolar traits that could serve as a basis for Jung’s 8-element typology.

---

¹ Translator’s note: D₈ here is the order-8 dihedral group — D₄ in the geometric naming convention.

² Translator’s note on restored glyphs: the quadra sets and the octochotomy sequences follow uniquely from the subgroup listings and the sign convention. The two example starting TIMs in the irrationals/rationals passage (rendered Se⁻ and Ti⁻) are not recoverable from the text layer — the original glyphs may differ, but any irrational/rational starting TIM makes the passage’s point identically.

The Socionic Compass as a Basis for Constructing Two 16-Element Groups of Bipolar Traits of TIMs with a Common 8-Element Subgroup

Translator’s note: this section’s pair notation (Joynton’s single-symbol notations for kindred, extinguishment, superego and dual pairs, now part of the socionics font) does not survive in the PDF’s text layer. Pairs are therefore written out in full braces, e.g. {Ne⁺, Ni⁺} for an extinguishment pair. Table 1.1 and Table 1.2 are images in the source; they are reconstructed below from the text’s own cross-references (see notes).

In constructing the group of JMTs we shall need the convention, fixed in Table 1.1, concerning the names of the bipolar traits of the information aspects. Using this table as an example, we shall also demonstrate the principle by which a binary operation is introduced on the set of bipolar traits of the IA.

Table 1.1. The group of bipolar traits of the information aspects used here (reconstructed):

№	Trait (pole 1 / pole 2)	Ne	Ni	Se	Si	Te	Ti	Fe	Fi
0	IA / not IA (existence)	+	+	+	+	+	+	+	+
1	abstract / involved	+	+	−	−	+	+	−	−
2	irrational / rational	+	+	+	+	−	−	−	−
3	internal / external	+	+	−	−	−	−	+	+
4	static / dynamic	+	−	+	−	−	+	−	+
5	α- / γ-values	+	−	−	+	−	+	+	−
6	extraverted / introverted	+	−	+	−	+	−	+	−
7	δ- / β-values	+	−	−	+	+	−	−	+

[Reconstruction notes: rows 1, 2, 4 are the “basis” rows set in bold in the original; rows 3, 5, 6, 7 are the products 1⊗2, 1⊗4, 2⊗4 and 3⊗4 respectively. Row assignments are fixed by the worked examples in the text (row 7 = δ/β-values as 3⊗4; α-values referenced back to the table). The verbal label of row 6 is not recoverable with certainty from the text layer; “extraverted / introverted” (black / white) is the standard verbalization of that section.]

The signs ”+” and ”−” in the table denote an IA’s membership in the first and second pole of the bipolar trait respectively. That the column “Ne” contains only “pluses” means that this IA has been taken as the “reference point” in the set under consideration. Row “0”, filled entirely with “pluses,” answers for the so-called identity section; the corresponding trait is called the trait of existence. In the group it plays the role of the identity (neutral) element. The traits “1”, “2” and “4”, set in bold, are adopted as the “basis” (the generating system).

Having specified sections “1” and “2” of the set of IA by ordered arrays of “pluses” and “minuses,” we obtain section “3” by a simple rule: if in one and the same column sections “1” and “2” have the same signs, then section “3” gets a ”+” in that column; if different, then a ”−” is to be placed in row “3”. In this way a binary operation can be introduced formally on the set of sections of any set [19, p. 149].

If we were considering the 4-element set of information macroaspects {N, S, T, F}, the only possible group of central sections would be given by the first four rows (“0”–“3”) of our Table 1.1. Of course, the number of columns in the “body” of the table would also be half as large: instead of the columns “Ne” and “Ni,” for example, there would be a single column “N”. Indeed, in rows “0”–“3” the signs in the columns “Ne” and “Ni” coincide anyway. One further change that would have to be made concerns the neutral element: the label of the row “IA / not IA” would have to be replaced by “macroaspects / not macroaspects.”

Dividing each macroaspect into two IA by the trait numbered 4 in Table 1.1, we automatically obtain, via the binary operation, three more sections (“5”–“7”). For this, row 4 must be “multiplied” by rows “1”–“3” respectively. For example, “3” ⊗ “4” = “7”; written out in full:

[the original displays the row-by-row multiplication here as part of the table graphic]

Let us explain the binary operation just introduced, on this concrete example, in a somewhat different way. To the first pole of the product belong those elements that belong simultaneously to the first poles of the factors, together with those that belong simultaneously to the second poles of the factors. There are exactly two information aspects that are simultaneously internal and static (Ne and Fi), and exactly two that are simultaneously external and dynamic (Si and Te). Hence to the first pole of the product belongs the quartet {Ne, Si, Te, Fi} (the δ-values); correspondingly, to the second pole belongs the quartet of the remaining IA, i.e. {Ni, Se, Ti, Fe} (the β-values).

Of course, such words for fixing the poles of the central section {Ne, Si, Te, Fi} / {Ni, Se, Ti, Fe} of the set of IA could be applied only after the concept of the values of the quadras had been introduced. Let us note in passing that the choice of words for verbalizing a particular section of a set is a separate task, one that cannot be solved by mathematical methods. If the poles of the dichotomies of a set are given by listing their elements, then constructing the whole group from a known generating system is a purely technical task — whereas disputes over the choice of the most apt terms for verbalizing a particular section can go on for a very long time.

For example, for this very section (δ/β-values) V.V. Gulenko proposed the variant “inertial / propulsive” [7]. For fixing certain other central sections of the set of IA he likewise uses terms differing from those of Table 1.1. The words he proposed for verbalizing the elements of the group under discussion could be applied, without special modification, to Jung’s 8-element typology, in which the psychological type is in effect specified by the information aspect of the first function. But the terms figuring in the two last rows of Table 1.1 could not be recognized as suitable for that purpose — because in Jung’s 8-element typology the very concept of a quadra could not, in principle, have arisen.

As has already been said, kindred types differ in the IA of the second function. But for a given IA of the first function there are only two variants for the IA of the second. The point is that the IA of the first two functions must differ by the trait irrational / rational but coincide by the trait static / dynamic. This rule made it possible to introduce the abbreviated symbolic notations of the TIMs using the superscripts ”+” (for the right) and ”−” (for the left). Let us note in passing that in the article [14] an attempt was made to explain the existence of this rule from the standpoint of L.M. Vekker’s Unified Theory of Psychic Processes [4]. We shall return to these explanations.

Now imagine that we have decided to build a table of bipolar traits of TIMs, making use of Table 1.1 for the IA. We double the number of columns in the “body” of the table on a very simple principle, whereby in place of, say, the column “Ne” there appear the columns “Ne⁺” and “Ne⁻“. The filling-in of these columns with “pluses” and “minuses” up to and including row “7” will be identical for kindred types, since those rows would be tied to the IA of the first function only. The verbalization of the sections will, of course, have to be adjusted slightly. But what is at issue now is the formal-mathematical way of obtaining the central sections of the socion belonging to one of the possible 16-element groups.

In row “8” the right TIMs would have to be given “pluses” and the left ones “minuses.” This row will correspond to the fourth element of the generating system of the group of order 16 (alongside the bipolar traits “1”, “2” and “4”). After that, we “multiply” row “8” by rows “1”–“7”, writing the results into rows “9”–“15” respectively. In this way the “body” of the whole table for one of the possible 16-element groups of central dichotomies of the socion is obtained. Once the “body” of the table is filled in, one will have to think about verbalizing the resulting sections of the socion (about how to label the rows).

Following the procedure described, one can obtain the group of bipolar traits of TIMs that is now called the group of JMTs. No great problems arose with the verbalization of the elements of this group. In the JMT group, besides the 8 elements belonging to the AR-trait group well known to socionists, there turned out to be 8 more, for which terms were proposed tied to the traits of the IA processed by the first or second functions of the TIMs [12].

Let us show how infographics based on the Socionic Compass can be used in constructing this same group of bipolar traits. To simplify the diagrams, we shall use conventional notations for kindred pairs and extinguishment pairs. For example, one symbol will denote the kindred pair {Ne⁺, Ne⁻}, and another the extinguishment pair {Ne⁺, Ni⁺}. The two necessary diagrams are presented in Fig. 1.7.

[Fig. 1.7. Using the Socionic Compass to introduce a generating system for the group of JMTs.]

Without dwelling yet on the principle behind the letter-and-digit notations for the bipolar traits, let us make several preliminary remarks. Recall that in verbalizing the sections of any set by bipolar traits one must agree on how the order of the poles is to be chosen — this is important for introducing the binary operation on the set of traits. In considering the set of IA, we took Ne as the “reference point.”

In Table 1.1 the corresponding column is filled entirely with “pluses.” In every row of the table, the “pluses” went to those IA belonging to the same pole of the respective bipolar trait as Ne. Although we have not exhibited the tables with the groups of bipolar traits of the macroaspects and of the TIMs, the descriptions of how they can be built presupposed that in them the “reference points” are N and Ne⁺ respectively.

In considering groups of bipolar traits of TIMs we shall continue to assume that the order of the poles is chosen so that the first is the pole corresponding to the ILE (Ne⁺). If the abbreviated symbolic notations of the TIMs are not shown explicitly on a diagram (see Fig. 1.7), one must orient oneself by the symbolic notations of the pairs or quartets to which they belong. This convention makes it possible, by means of call-outs on the diagrams, to give compact labels to the lines symbolizing the sections of the socion.

Let us see, for example, how the characteristics of the ESE (Fe⁻) are represented in Fig. 1.7. It is easy to discover that this TIM belongs to the first pole of the trait B1 (like Ne⁺), and to the second pole of the traits A0, B0 and I3 (unlike Ne⁺). Hence Fe⁻ should be characterized by the words: internal by the first function, dynamic, involved by the first function, left. If we are familiar with the bipolar traits of the IA given in Table 1.1 and know how the TIMs of one kindred pair are separated by the trait right / left, we can verify the correctness of the result obtained by finding all these characteristics directly in the TIM’s abbreviated symbolic notation.

The left diagram of Fig. 1.7 demonstrates well the fact that the triple of bipolar traits {B0, B1, A0} can serve as a “basis” of the 8-element typology of kindred pairs. Indeed, the lines corresponding to these traits (two straight lines and a circle) separate all the elements from one another. The right diagram indicates that the triple {B0, B1, I3} is a “basis” of the 8-element typology of extinguishment pairs. A specific TIM is fully determined by specifying the two kinds of pairs to which it belongs. Consequently, the quadruple of bipolar traits {B0, B1, A0, I3} can serve as a “basis” for the typology of the socionic types. Knowing to which poles of these four traits a given TIM belongs makes it possible, in principle, to determine to which pole of any (not necessarily verbalized!) dichotomy of the socion it belongs.

If one takes into account exclusively the central sections of the socion verbalized in the AR-trait and JMT groups, the number of additional ones (beyond the four indicated) turns out to be 19. Of these, 5 enter the common subgroup of the two groups, 6 only the JMT group, and 8 only the AR-trait group. The calculation is very simple. Not counting the identity section, the number of central dichotomies of the socion verbalized in these groups is 23 [10] (15 in each, but 7 in common). And in the quadruple {B0, B1, A0, I3} two traits are common (A0 and I3), while the other two belong only to the JMT group.

Having settled on one of the variants of the generating system of the JMT group, the remaining 11 sections of this group (not counting the identity one) could be obtained by means of a table analogous to Table 1.1, allotting the “basis” sections {B0, B1, A0, I3} the rows “1”, “2”, “4” and “8” (not necessarily in that order). The remaining rows are filled in automatically by the rule stated earlier, which realizes the binary operation adopted on the set of bipolar traits (i.e. the appearance of the “body” of the table, under this approach to filling it in, does not depend on which suitable quadruple of bipolar traits is taken as the “basis”).

Let us note that, in introducing the binary operation in question, G.R. Reinin explained it using a diagram similar to those given in Fig. 1.7. The corresponding figure in his book is called “Partition of the socion by two orthogonal sections” [19, p. 147]. There, too, there is a circle, which in effect represented the identity section, separating the elements of the socion (the set under study) from all other objects. In the diagrams presented in Fig. 1.7, by contrast, the circle has the meaning of yet another central section, which together with the sections represented by straight lines enters the “basis” of the 8-element typology of pairs of TIMs (kindred pairs on the left diagram, extinguishment pairs on the right). Of the identity section one can say that on our diagrams it is represented by the outline of the “cross” as a constituent part of the Socionic Compass: all the notations of the elements of the sets under consideration lie inside that outline.

How does one find the “product” of the sections B0 and B1? Can this be done without compiling a table of “pluses” and “minuses”? Let us use the left diagram of Fig. 1.7. As we have said, to the first pole of the product belong those elements that belong simultaneously to the first poles of the factors, together with those belonging simultaneously to the second poles of the factors. From the diagram it is easy to determine that the TIMs belonging simultaneously to the first poles of the factors are those in the kindred pairs {Ne⁺, Ne⁻} and {Ni⁺, Ni⁻}, and to the second poles, those in the pairs {Se⁺, Se⁻} and {Si⁺, Si⁻}. Consequently, to the first pole of the product belong all the irrationals, the symbols of whose pairs lie on the vertical axis of symmetry of the “cross.” Hence the sought product is the trait irrationals / rationals (B0 ⊗ B1 = I1).

Using the same diagram, let us find the “product” of the sections B0 and A0. To the first poles of these sections simultaneously belong the pairs {Ne⁺, Ne⁻} and {Ti⁺, Ti⁻}, and to the second poles {Si⁺, Si⁻} and {Fe⁺, Fe⁻}. If one attends to the IA symbols of the first functions of the TIMs in these kindred pairs, one recognizes in the quartet {Ne, Ti, Si, Fe} the α-values (see Table 1.1). Hence the section corresponding to the product can be verbalized as α / γ — 1, where the “1” is to be read “values by the first function.” In abbreviated form: B0 ⊗ A0 = AB0.

Let us consider a case harder to verbalize, using the right diagram, on which the abbreviated symbolic notations of the extinguishment pairs figure. Let us find the product of the sections I3 and B1. Proceeding by the same algorithm as in the two previous cases, we find that to the first pole of the product must be assigned the TIMs belonging to the extinguishment pairs of the quartet {N⁺, F⁺, S⁻, T⁻} — that is, {Ne⁺, Ni⁺}, {Fe⁺, Fi⁺}, {Se⁻, Si⁻}, {Te⁻, Ti⁻}. How is the corresponding section to be verbalized? Let us write out all the TIMs of this quartet of extinguishment pairs in the unfolded symbolic form: {(Ne, Ti), (Ni, Te), (Fe, Ni), (Fi, Ne), (Se, Ti), (Si, Te), (Te, Ni), (Ti, Ne)}. Now attend to the IA processed by the second functions of the TIMs of the resulting octet. They belong to the first pole of the trait abstract / involved that was used for the set of IA (see Table 1.1). This gives grounds to propose, for the resulting section of the set, the name abstract / involved — 2, where the “2” is to be read “by the second function.” In abbreviated form: B1 ⊗ I3 = B2.

Let us turn once more to the right diagram and try to find the section I2, denoting thus the product I1 ⊗ I3. We already know that I1 = irrationals / rationals. To the first pole of this trait correspond the symbols lying on the vertical axis of symmetry of the “cross”; the remaining symbols (corresponding to the rational part of the socion) lie on the horizontal axis. Directly from Fig. 1.7 one can read that I3 = right / left and see that on the right diagram this bipolar trait is represented by a dividing line in the form of a circle. The general rule for multiplying bipolar traits tells us that to the first pole of I2 belong the right irrationals (first poles of I3 and I1) and the left rationals (second poles of I3 and I1). The TIMs of interest belong to the following quartet of extinguishment pairs: {N⁺, S⁺, T⁻, F⁻} — that is, {Ne⁺, Ni⁺}, {Se⁺, Si⁺}, {Te⁻, Ti⁻}, {Fe⁻, Fi⁻}. What suitable word unites them?

One may notice that the TIMs of these pairs belong either to the Club of Scientists ({N⁺, T⁻}) or to the Club of Socials ({S⁺, F⁻}). Consequently, they are democrats. We conclude: I2 = democrats / aristocrats. Note that these same TIMs could also be distributed into the quartets {Ne⁺, Si⁺, Fe⁻, Ti⁻} and {Se⁺, Ni⁺, Te⁻, Fi⁻}. As we see, these are the quadras α and γ. It is no accident that in socionics these quadras are called democratic, in contrast to the aristocratic ones (β and δ).

The letter-and-digit notations for the ARTs and JMTs were introduced by us precisely so that the binary operation in these groups could be carried out quite simply, without resorting to cumbersome tables or diagrams. Since these groups share a common 8-element subgroup, it made sense to choose “bases” (generating systems) for them differing in only one element. The generating system of the common subgroup acquired in these notations the form {I1, I2, A0}, abbreviating the unfolded record {irrationals / rationals, democrats / aristocrats, statics / dynamics}. To obtain the “basis” of the JMT group, this triple was supplemented by the trait B0 (abstract / involved — 1). In the AR-trait group, the bipolar trait C0 (judicious / decisive) was taken as the fourth element of the “basis” [18].

It proved possible to introduce the letter-and-digit notations of the bipolar traits in such a way that, in carrying out the binary operation, the digital and letter parts transform independently of each other. Two isomorphic groups of order 4 resulted, representable as C2×C2: {0, 1, 2, 3} and {I, A, B, AB}. Their identity (neutral) elements are “0” and “I” respectively. Recall that we have already met such groups: the right graph of Fig. 1.6 is their Cayley graph. Such are 13 of the 15 fourth-order subgroups of the group of operators of the classical IR; the group of central bipolar traits of the information macroaspects is isomorphic to them as well.

Let us use the products of JMTs we have analyzed in detail to demonstrate the independent execution of the binary operation in the digital and letter parts of their notations:

• B ⊗ B = I, 0 ⊗ 1 = 1 ⟹ B0 ⊗ B1 = I1;
• B ⊗ A = AB, 0 ⊗ 0 = 0 ⟹ B0 ⊗ A0 = AB0;
• B ⊗ I = B, 1 ⊗ 3 = 2 ⟹ B1 ⊗ I3 = B2;
• I ⊗ I = I, 1 ⊗ 3 = 2 ⟹ I1 ⊗ I3 = I2.

As for the AR-trait group, its letter parts are taken from the quadruple {I, A, C, AC}. Let us show how one of the possible “bases” of this group is obtained using the Socionic Compass. In this case its intuitive, sensory, ethical and logical poles are represented differently (Fig. 1.8).

[Fig. 1.8. Using the Socionic Compass to introduce a generating system for the AR-trait group.]

Note that the “basis” of the ARTs presented in this figure differs from the so-called “Jung basis,” which G.R. Reinin used in constructing the group, in just one element: in place of I1 (irrationals / rationals), A0 (statics / dynamics) is used here. But I1 is easily obtained as the product A0 ⊗ A1. In Fig. 1.8 it is also easy to see the division of the socion by the trait I2 (democrats / aristocrats), since it is obtained as the product AC0 ⊗ AC2.

Indeed, as we know, the democrats comprise the intuitives who are at the same time logicians ({Ne⁺, Ti⁻, Ni⁺, Te⁻}, the Club of Scientists) and the sensorics who are at the same time ethicists ({Se⁺, Fe⁻, Si⁺, Fi⁻}, the Club of Socials). To the aristocratic half of the socion belong, correspondingly, the intuitives who are at the same time ethicists ({Ne⁻, Fe⁺, Ni⁻, Fi⁺}, the Club of Humanitarians) and the sensorics who are at the same time logicians ({Se⁻, Te⁺, Si⁻, Ti⁺}, the Club of Managers).

Thus by means of two orthogonal dichotomies (AC0 and AC2) one can obtain the tetrachotomy of the socion into the quartets of TIMs that came to be called the professional clubs. Exactly the same partition is obtained by means of the 4-element subgroup {1, c, m, cm} of the group of operators of the classical IR.

Recall that we have already spoken of the quadras as the quartets of TIMs obtained under the tetrachotomy of the socion by means of the subgroup {1, -c, m, -cm}. If any club can be represented as two extinguishment pairs (c), any quadra is representable as two dual pairs (-c):

{{Ne⁺, Si⁺}, {Fe⁻, Ti⁻}} — α; {{Fe⁺, Ti⁺}, {Se⁻, Ni⁻}} — β; {{Se⁺, Ni⁺}, {Te⁻, Fi⁻}} — γ; {{Te⁺, Fi⁺}, {Ne⁻, Si⁻}} — δ.

Let us note in passing that such notations for the dual pairs first appeared in the article [17]. Besides the notations for dual pairs and extinguishment pairs we shall also need notations for the superego pairs: {Ne⁺, Se⁺}, {Ni⁺, Si⁺}, {Te⁻, Fe⁻}, and so on.¹ Such notations were proposed in private correspondence by Andrew Joynton. Now, thanks to the efforts of Olga Bogdanovna Karpenko, these symbols already form part of the socionic font, which it is convenient to use when writing works on socionics.

If I2 = {α, γ} / {β, δ}, then which bipolar traits correspond to the dichotomies {α, δ} / {β, γ} and {α, β} / {γ, δ}? The answer can be found with the help of the left diagram of Fig. 1.8, which has dividing lines corresponding to the traits of the triple {AC0, AC2, A0}. It is therefore easy to discover that the sought dichotomies are the following products respectively:

A0 ⊗ AC0 = C0 (judicious / decisive), A0 ⊗ AC2 = C2 (merry / serious).

As was to be expected, the relation C0 ⊗ C2 = I2 holds. Two further sections of the socion whose product is I2 can be found with the help of the right diagram of Fig. 1.8, which has dividing lines corresponding to the traits of the triple {AC0, AC2, A1}:

A1 ⊗ AC0 = C1 (carefree / farsighted), A1 ⊗ AC2 = C3 (yielding / obstinate).

Making use of the fact that, in the variants of the Socionic Compass presented in Fig. 1.8, the symbols of the irrationals lie — though not directly on the vertical axis of symmetry of the “cross” — close to it, and the symbols of the rationals close to the horizontal one, it is easy to find two more sections:

I1 ⊗ AC0 = AC1 (tacticians / strategists), I1 ⊗ AC2 = AC3 (constructivists / emotivists).

The section A0 ⊗ I3 = A3 (positivists / negativists) can be found using the left diagram of Fig. 1.8, where the circle separates the statics from the dynamics — and the division into right and left is easily tracked by the “pluses” and “minuses” used as superscripts in the TIMs’ abbreviated symbolic notations. The right diagram of Fig. 1.8, on which the circle separates the extraverts from the introverts, will help find the section A1 ⊗ I3 = A2 (questims / declatims).

All the central sections of the socion verbalized in the AR-trait and JMT groups can be represented as sections of 8-element sets of pairs of TIMs. We shall consider three kinds of pairs: superego (-1), extinguishment (c) and dual (-c).

We begin with the elements common to the AR-trait and JMT groups. The letter part of the formalized notation in this case is either “I” or “A”. In the former case any of the three kinds of pairs of TIMs may be used to display the sections. The trait of existence, responsible for the identity section and playing the role of the identity (neutral) element, is denoted I0; to its first pole belong all 16 TIMs (the whole socion). In the remaining cases each pole of the bipolar trait contains 8 TIMs, or 4 pairs. Using the pairs of each of the three kinds indicated, we obtain the following result:

I1 = irrationals / rationals = {Ne⁺,Se⁺}, {Ni⁺,Si⁺}, {Ne⁻,Se⁻}, {Ni⁻,Si⁻} / {Te⁺,Fe⁺}, {Ti⁺,Fi⁺}, {Te⁻,Fe⁻}, {Ti⁻,Fi⁻} = {Ne⁺,Ni⁺}, {Se⁺,Si⁺}, {Ne⁻,Ni⁻}, {Se⁻,Si⁻} / {Te⁺,Ti⁺}, {Fe⁺,Fi⁺}, {Te⁻,Ti⁻}, {Fe⁻,Fi⁻} = {Ne⁺,Si⁺}, {Se⁺,Ni⁺}, {Ne⁻,Si⁻}, {Se⁻,Ni⁻} / {Te⁺,Fi⁺}, {Fe⁺,Ti⁺}, {Te⁻,Fi⁻}, {Fe⁻,Ti⁻};

I2 = democrats / aristocrats = {Ne⁺,Se⁺}, {Ni⁺,Si⁺}, {Te⁻,Fe⁻}, {Ti⁻,Fi⁻} / {Ne⁻,Se⁻}, {Ni⁻,Si⁻}, {Te⁺,Fe⁺}, {Ti⁺,Fi⁺} = {Ne⁺,Ni⁺}, {Se⁺,Si⁺}, {Te⁻,Ti⁻}, {Fe⁻,Fi⁻} / {Ne⁻,Ni⁻}, {Se⁻,Si⁻}, {Te⁺,Ti⁺}, {Fe⁺,Fi⁺} = {Ne⁺,Si⁺}, {Se⁺,Ni⁺}, {Fe⁻,Ti⁻}, {Te⁻,Fi⁻} / {Ne⁻,Si⁻}, {Se⁻,Ni⁻}, {Fe⁺,Ti⁺}, {Te⁺,Fi⁺};

I3 = right / left = {Ne⁺,Se⁺}, {Ni⁺,Si⁺}, {Te⁺,Fe⁺}, {Ti⁺,Fi⁺} / {Ne⁻,Se⁻}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻}, {Ti⁻,Fi⁻} = {Ne⁺,Ni⁺}, {Se⁺,Si⁺}, {Te⁺,Ti⁺}, {Fe⁺,Fi⁺} / {Ne⁻,Ni⁻}, {Se⁻,Si⁻}, {Te⁻,Ti⁻}, {Fe⁻,Fi⁻} = {Ne⁺,Si⁺}, {Se⁺,Ni⁺}, {Te⁺,Fi⁺}, {Fe⁺,Ti⁺} / {Ne⁻,Si⁻}, {Se⁻,Ni⁻}, {Te⁻,Fi⁻}, {Fe⁻,Ti⁻}.

The second quartet of traits belonging to the common 8-element subgroup (with “A” as the letter part of the formalized notation) cannot be represented using extinguishment (c) or dual (-c) pairs, but can be using superego pairs (-1):

A0 = statics / dynamics = {Ne⁺,Se⁺}, {Ti⁺,Fi⁺}, {Ne⁻,Se⁻}, {Ti⁻,Fi⁻} / {Ni⁺,Si⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻};

A1 = extraverts / introverts = {Ne⁺,Se⁺}, {Te⁺,Fe⁺}, {Ne⁻,Se⁻}, {Te⁻,Fe⁻} / {Ni⁺,Si⁺}, {Ti⁺,Fi⁺}, {Ni⁻,Si⁻}, {Ti⁻,Fi⁻};

A2 = questims / declatims = {Ne⁺,Se⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Ti⁻,Fi⁻} / {Ni⁺,Si⁺}, {Ti⁺,Fi⁺}, {Ne⁻,Se⁻}, {Te⁻,Fe⁻};

A3 = positivists / negativists = {Ne⁺,Se⁺}, {Ti⁺,Fi⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻} / {Ni⁺,Si⁺}, {Te⁺,Fe⁺}, {Ne⁻,Se⁻}, {Ti⁻,Fi⁻}.

We pass to the bipolar traits belonging to the JMT group but not the AR-trait group. The “B”-traits can be expressed using extinguishment pairs (c), but superego (-1) or dual (-c) pairs will not do in this case:

B0 = abstract-1 / involved-1 = {Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Te⁺,Ti⁺}, {Te⁻,Ti⁻} / {Se⁺,Si⁺}, {Se⁻,Si⁻}, {Fe⁺,Fi⁺}, {Fe⁻,Fi⁻};

B1 = internal-1 / external-1 = {Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Fe⁺,Fi⁺}, {Fe⁻,Fi⁻} / {Se⁺,Si⁺}, {Se⁻,Si⁻}, {Te⁺,Ti⁺}, {Te⁻,Ti⁻};

B2 = abstract-2 / involved-2 = {Ne⁺,Ni⁺}, {Fe⁺,Fi⁺}, {Se⁻,Si⁻}, {Te⁻,Ti⁻} / {Se⁺,Si⁺}, {Te⁺,Ti⁺}, {Ne⁻,Ni⁻}, {Fe⁻,Fi⁻};

B3 = external-2 / internal-2 = {Ne⁺,Ni⁺}, {Te⁺,Ti⁺}, {Se⁻,Si⁻}, {Fe⁻,Fi⁻} / {Se⁺,Si⁺}, {Fe⁺,Fi⁺}, {Ne⁻,Ni⁻}, {Te⁻,Ti⁻}.

If the first two traits of this quartet are well reflected in the TIMs’ abbreviated symbolic notations, then to see the last two (B2 and B3) behind such notations one must learn to reconstruct quickly, in one’s head, the IA of the second function from the IA symbol of the first function and the sign used as superscript. We have already discussed how this can be done by picturing the Socionic Compass in the form shown in Figs. 1.3 and 1.4. But it is useful also to remember the link between the signs and the professional clubs. For example, intuition with a “plus” is the kind found among the Scientists — hence in that case logic will stand in second place, and logic is abstract and external. And this does not depend on whether we are speaking of the ILE (Ne⁺) or the ILI (Ni⁺).

The “AB”-traits are expressed by means of dual (-c) pairs, but superego (-1) or extinguishment (c) pairs will not do here:

AB0 (α / γ — 1) = {Ne⁺,Si⁺}, {Fe⁻,Ti⁻}, {Ne⁻,Si⁻}, {Fe⁺,Ti⁺} / {Se⁺,Ni⁺}, {Te⁻,Fi⁻}, {Se⁻,Ni⁻}, {Te⁺,Fi⁺};

AB1 (δ / β — 1) = {Ne⁺,Si⁺}, {Ne⁻,Si⁻}, {Te⁺,Fi⁺}, {Te⁻,Fi⁻} / {Fe⁺,Ti⁺}, {Fe⁻,Ti⁻}, {Se⁺,Ni⁺}, {Se⁻,Ni⁻};

AB2 (α / γ — 2) = {Ne⁺,Si⁺}, {Te⁺,Fi⁺}, {Fe⁻,Ti⁻}, {Se⁻,Ni⁻} / {Se⁺,Ni⁺}, {Fe⁺,Ti⁺}, {Ne⁻,Si⁻}, {Te⁻,Fi⁻};

AB3 (β / δ — 2) = {Ne⁺,Si⁺}, {Fe⁺,Ti⁺}, {Se⁻,Ni⁻}, {Te⁻,Fi⁻} / {Se⁺,Ni⁺}, {Te⁺,Fi⁺}, {Ne⁻,Si⁻}, {Fe⁻,Ti⁻}.

In this quartet, to see the poles of the traits AB2 and AB3 in the TIMs’ abbreviated symbolic notations, one must again be able to “compute” the IA of the second function quickly. Besides the methods already considered, it is useful to remember that in the democratic quadras it is the irrationals who are right, and in the aristocratic ones the rationals. This is well seen in Table 1.2, where the quadras “begin” with the right dual pairs.

Table 1.2 (reconstructed):

Quadra	Right dual pair	Left dual pair
α	{Ne⁺, Si⁺}	{Fe⁻, Ti⁻}
β	{Fe⁺, Ti⁺}	{Se⁻, Ni⁻}
γ	{Se⁺, Ni⁺}	{Te⁻, Fi⁻}
δ	{Te⁺, Fi⁺}	{Ne⁻, Si⁻}

It must be said that this order of the dual pairs within the quadras is the one used most often in socionic publications, though there are works whose authors propose other variants (see e.g. [9], [21]). If one adopts the order of dual pairs within the quadras as in Table 1.2, and represents the socion as a closed structure in which the α-quadra follows again after the δ-quadra, one obtains a simple rule: for the right TIMs, second place is occupied by an IA belonging to the values not only of their own quadra but also of the next, while for the left TIMs — of the preceding one.

Let us note once more that the names of the 8 last traits (of the quartets “B” and “AB”), belonging only to the JMT group and not the AR-trait group, were proposed with a direct tie to the traits of the IA processed by the first or second functions of the TIMs. Their poles are easy to name once the TIM in question is known. But filling them with a psychological meaning that would help in determining the socionic type of a concrete person is a separate and far-from-simple task. Let us only draw attention to the order of the poles in the trait pairs {B1, B3} and {AB1, AB3}: to avoid confusion, one must always remember that the ILE (Ne⁺ = (Ne, Ti)) is adopted as the “reference point.”

We pass to the bipolar traits belonging to the AR-trait group but not the JMT group. The “C”-traits of the AR group, like the “AB”-traits of the JMT group, are likewise expressed by means of dual (-c) pairs:

C0 = judicious / decisive = {Ne⁺,Si⁺}, {Fe⁻,Ti⁻}, {Te⁺,Fi⁺}, {Ne⁻,Si⁻} / {Fe⁺,Ti⁺}, {Se⁻,Ni⁻}, {Se⁺,Ni⁺}, {Te⁻,Fi⁻};

C1 = carefree / farsighted = {Ne⁺,Si⁺}, {Fe⁺,Ti⁺}, {Te⁻,Fi⁻}, {Ne⁻,Si⁻} / {Fe⁻,Ti⁻}, {Se⁻,Ni⁻}, {Se⁺,Ni⁺}, {Te⁺,Fi⁺};

C2 = merry / serious = {Ne⁺,Si⁺}, {Fe⁻,Ti⁻}, {Fe⁺,Ti⁺}, {Se⁻,Ni⁻} / {Se⁺,Ni⁺}, {Te⁻,Fi⁻}, {Te⁺,Fi⁺}, {Ne⁻,Si⁻};

C3 = yielding / obstinate = {Ne⁺,Si⁺}, {Te⁺,Fi⁺}, {Te⁻,Fi⁻}, {Se⁻,Ni⁻} / {Fe⁻,Ti⁻}, {Fe⁺,Ti⁺}, {Se⁺,Ni⁺}, {Ne⁻,Si⁻}.

At the same time, the “AC”-traits are expressed by means of extinguishment pairs (c):

AC0 = intuitives / sensorics = {Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Te⁻,Ti⁻}, {Fe⁺,Fi⁺} / {Se⁺,Si⁺}, {Se⁻,Si⁻}, {Te⁺,Ti⁺}, {Fe⁻,Fi⁻};

AC1 = tacticians / strategists = {Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Te⁺,Ti⁺}, {Fe⁻,Fi⁻} / {Se⁺,Si⁺}, {Se⁻,Si⁻}, {Te⁻,Ti⁻}, {Fe⁺,Fi⁺};

AC2 = logicians / ethicists = {Ne⁺,Ni⁺}, {Se⁻,Si⁻}, {Te⁺,Ti⁺}, {Te⁻,Ti⁻} / {Ne⁻,Ni⁻}, {Se⁺,Si⁺}, {Fe⁺,Fi⁺}, {Fe⁻,Fi⁻};

AC3 = constructivists / emotivists = {Ne⁺,Ni⁺}, {Se⁻,Si⁻}, {Fe⁺,Fi⁺}, {Fe⁻,Fi⁻} / {Ne⁻,Ni⁻}, {Se⁺,Si⁺}, {Te⁺,Ti⁺}, {Te⁻,Ti⁻}.

And so, after explaining how all the elements of the two 16-element groups of bipolar traits of TIMs with a common 8-element subgroup can be obtained by means of the various representations of the Socionic Compass given in Figs. 1.7 and 1.8, we have explicitly written out all the corresponding sections of the socion. Admittedly, in the enumeration we used the symbols not of individual TIMs but of their pairs — the pairs themselves being formed, depending on the bipolar trait under consideration, by means of one operator of symmetric IR from the triple {superego (-1), extinguishment (c), dual (-c)}.

Note that by means of this triple the socion is divided into four parts in exactly the same way as by any pair from the triple of mutually dependent traits {I1 (irrationals / rationals), I2 (democrats / aristocrats), I3 (right / left)}. If the triple of operators indicated is supplemented by the identity operator (1), one obtains the centre of the 16-element non-commutative group of operators of the classical IR (i.e. the subgroup gathering all the operators that commute with the elements of the whole group). If the triple of bipolar traits indicated is supplemented by the trait of existence (I0), one obtains a group of order 4 that belongs both to the AR-trait group and to the JMT group.

Since the groups of bipolar traits are commutative, every subgroup is normal (in the mathematical sense of the word), which allows the group to be decomposed into cosets with respect to any subgroup — the left and right cosets then coincide. Let us decompose the JMT and AR-trait groups by the subgroup {I0, I1, I2, I3}, using the formalized notations.

JMT group: 1) {I0, I1, I2, I3}, 2) {A0, A1, A2, A3}, 3) {B0, B1, B2, B3}, 4) {AB0, AB1, AB2, AB3}.

AR-trait group: 1) {I0, I1, I2, I3}, 2) {A0, A1, A2, A3}, 3) {C0, C1, C2, C3}, 4) {AC0, AC1, AC2, AC3}.

The cosets with respect to a normal subgroup themselves form a group [5, p. 169]. The elements of the groups {I, A, B, AB} and {I, A, C, AC} of which we spoke earlier are precisely the cosets of the JMT and AR-trait groups with respect to the subgroup I. As for the lower digital indices used in our formalized notations of the bipolar traits, they formed the group {0, 1, 2, 3} precisely because the quartet of traits {I0, I1, I2, I3} is a group (in the mathematical sense of the word).

The 16-element AR-trait and JMT groups themselves are representable as the direct product of four of their cyclic subgroups of order 2. For example, the JMT group can be represented thus: {I0, I1}×{I0, I2}×{I0, A0}×{I0, B0}. But it can also be represented thus: {I0, I1, I2, I3}×{I0, A0, B0, AB0}. The latter representation, as the direct product of the two 4-element subgroups indicated, confirms the rule of independent multiplication of the letter and digital parts of the proposed formalized notations of the bipolar traits in the JMT group. In the AR-trait group the situation is analogous.

---

¹ Translator’s note: the specific superego-pair examples shown in the original are not recoverable from the text layer; representative pairs are given. All octet listings in this section were reconstructed from the trait definitions and verified against the group’s multiplication rules (letter and digit parts computed independently); each pole was checked to contain the correct reference elements.

The Socionic Compass and the Models “A” of the TIMs

In the preceding sections of this chapter the TIMs were represented by their abbreviated symbolic notations. This was quite sufficient for building the Socionic Compass, on whose poles were placed the symbols of the TIMs united by the macroaspect to which the IA of their first functions belong. The resulting new object of infographics, reminiscent of Jung’s Compass, allowed us to obtain, by simple and entirely formal means, three 16-element groups (the group of operators of the classical IR and two groups of bipolar traits of TIMs). But for a detailed description of the types and the intertype relations one would need to know which IA occupy not only the first two functions but the six remaining ones as well. Let us see how the Socionic Compass is connected with the Models “A” of the TIMs.

The signs in the abbreviated symbolic notations of the TIMs have so far been associated only with the bipolar trait right / left, used to distinguish kindred types in the passage from Jung’s 8-element typology to the 16-element socionic typology. But the signs can also be used to divide each IA into two microaspects, depending on the symbol of which other IA stands in the same horizontal block of the Model “A” of the specific TIM (see e.g. [3], [20]). Let us note in passing that there also exists another approach to the placement of signs in the Models “A” of the socionic types (see e.g. [8], [22]), which we shall not consider. The horizontal blocks of the Models “A” have special names. The Ego block, uniting the 1st and 2nd functions, has already been discussed. The remaining six functions are distributed over the horizontal blocks as follows: Superego (3rd and 4th), Superid (5th and 6th), Id (7th and 8th).

Under the approach to sign placement oriented to the combination of IA in the horizontal blocks of the Models “A,” a simple rule operates: among democrats the sign ”+” goes to the irrational aspects, and among aristocrats to the rational ones. If we now return to the “unfolded” variant of the Socionic Compass presented in Fig. 1.4, the Models “A” of all 16 TIMs can be seen in it.

The models of the democrats are more clearly visible if one keeps only the symbols of the irrational IA with the sign ”+” and of the rational ones with the sign ”−“. Fig. 1.9 shows, on the example of the SEI, how the numbers of the socionic functions are to be placed in the case of the right democrats. The mental ring (functions 1–4) is represented by the symbols lying on that circle of the Socionic Compass where the abbreviated symbolic notation of the given type is located — a notation which must now be understood as denoting the IA of the first function with the aspect of its Ego-block neighbour taken into account.

[Fig. 1.9. Using the Socionic Compass to build the Model “A” of the SEI as a representative of the right democrats (for each TIM the numbering of the functions begins from the corresponding cell).]

The direction in which the numbers increase from “1” to “4” is shown by an arrow (clockwise). The vital ring of the SEI (functions 5–8) is the responsibility of the other “democratic” circle of the Socionic Compass. Attention must be paid to where the cell of the fifth function lies relative to the cell of the third function, whose IA symbol differs only in colour. The increase of the numbers from “5” to “8” is given by the same arrows.

For the left democrats the direction of the arrows must be reversed. As an example, Fig. 1.10 takes the ESE. As we see, the position of the IA symbols has not changed; one simply has to count the numbers from a different cell. This cell, too, turned out to lie on the circle of smaller radius.

[Fig. 1.10. Using the Socionic Compass to build the Model “A” of the ESE as a representative of the left democrats.]

This is because from the outset we allotted, in the Socionic Compass (see Fig. 1.4), the two outer circles to the static TIMs and the two inner ones to the dynamic. When we try to see the Model “A” of a specific TIM in the Socionic Compass, only two circles remain in the field of our attention. The circle of larger radius corresponds to the static ring of the Model “A.” Since the SEI and the ESE are dynamics, their mental rings correspond to the circles of smaller radius. But this does not fundamentally affect the placement of the function numbers. The passage from one circle to the other (in going from the mental ring to the vital one) always happens in the same way: one must orient oneself by the position of the cell of the fifth function relative to the cell of the third.

Figs. 1.11 and 1.12 give examples for the aristocratic types.

[Fig. 1.11. Using the Socionic Compass to build the Model “A” of the LSI as a representative of the right aristocrats.]

[Fig. 1.12. Using the Socionic Compass to build the Model “A” of the SLE as a representative of the left aristocrats.]

What, then, has changed in the case of the aristocratic TIMs? The difference consists only in the fact that in the Socionic Compass one must use the “aristocratic” circles. In Fig. 1.11 the LSI is a representative of the right aristocrats, and in Fig. 1.12 the SLE represents the left aristocrats.

We have discussed how the Model “A” of any TIM can be seen in the Socionic Compass. For this we had to attend to which poles of the bipolar traits democrats / aristocrats and right / left the specific type belongs. On this depended which two of the four circles to concentrate one’s attention on, and also in which direction (clockwise or counterclockwise) the numbers of the socionic functions would increase.

Let us note that in the Socionic Compass one can see not only the 8-component Models “A” of all the TIMs but also the 16-component Models “B” [2]. In that case one need not exclude from consideration the two circles bearing the symbols of the TIMs belonging to one of the poles of the trait democrats / aristocrats. Of course, these symbols must now be understood to stand not for TIMs but for IA with their signs.

The ninth number should be assigned to the cell containing the symbol that differs from the symbol of the first cell only in sign. The direction of increase of the numbers from “9” to “16” is opposite to the direction of increase from “1” to “8”. This is because the shadow component of the 16-component Model “B” (numbers 9–16) formally coincides with the principal component (numbers 1–8) of the kindred TIM — and kindred TIMs differ by the trait right / left, which is precisely what determines the direction of number increase that interests us.

Most often the Models “A” of the TIMs are used in the form presented in the lower-left corner of Figs. 1.9–1.12. In some cases other variants of these same models prove useful, including spatial (3D) ones. In the variant reflected in the figures mentioned, the signs ”+” and ”−” are usually not written, since in that variant it is in any case clearly visible how the functions processing the corresponding IA are united into horizontal blocks. In the 16-element Models “B,” however, one cannot, of course, do without the signs.

Let us note once again that in this book we use precisely that approach to the placement of signs under which their choice is determined by which information aspects are united into the horizontal blocks of the Models “A,” which have the form of 4×2 tables. Under this approach one may speak of microaspects of the information flow, which differ in democratic and aristocratic blocks. The result is that among the democrats all the horizontal blocks in the Models “A” are democratic, and among the aristocrats aristocratic. But there are adherents of another approach which, compared with the one we have considered, leads to opposite signs in the vital rings (functions 5–8). This fact must be borne in mind when reading the socionics literature.

Conclusion to Chapter 1

In this chapter a new element of socionic infographics has been examined, whose prototype was Jung’s Compass. It is proposed to call this new element the Socionic Compass. Unlike the prototype, in which the four poles were tied to the four psychic functions figuring in C.G. Jung’s theory, each pole of the Socionic Compass is put into correspondence with a definite quartet of TIMs. The uniting into a quartet proceeds on the principle of a common macroaspect, to which belong the IA processed by the first functions of the types entering it.

The various variants of the Socionic Compass have made it possible to present, in a relatively simple and visual form, the structure of three 16-element groups built in the course of the development of the mathematical methods of socionics: the group of operators of the classical intertype relations and two groups of bipolar traits of TIMs with a common 8-element subgroup.

The elements of these groups were represented by their formalized notations, in which their group-theoretic (in the mathematical sense of the word) properties found reflection. The proposed system of notation allows one to carry out the binary operations in the groups indicated by following simple formalized rules, without resorting each time to cumbersome tables, diagrams and matrices. In this chapter, tables and diagrams are used to explain these rules, which in their final form are quite simple.

In addition, it was shown how the Models “A” of all 16 types of information metabolism can be seen in the “unfolded” variant of the Socionic Compass. Having understood to which pole of the bipolar trait democrats / aristocrats a given TIM belongs, one must concentrate one’s attention on the two corresponding circles of the four present in the “unfolded” variant of the Socionic Compass. After that, the 8 socionic symbols remaining in the field of attention should be understood as symbolic notations of the microaspects of the information flow, which for the given TIM form either democratic or aristocratic blocks. From the resulting four blocks the Model “A” of the specific type is formed by a simple rule.

Let us note that in the present chapter, without special explanation, the operators of the classical intertype relations were replaced by operators of transformations of the abbreviated symbolic notations of the TIMs. It ought to be proved that the group of operators of these transformations is indeed isomorphic to the group of operators of the classical IR. This will be done in the next chapter, for the various representations of the Models “A” of the TIMs.

References for Chapter 1

1. Augustinavichiute A. Sotsionika [Socionics]. — Moscow: Chernaya belka, 2008. — 568 pp.
2. Bukalov A.V. The 16-component model of the TIM and the socion // Sotsionika, mentologiya i psikhologiya lichnosti [Socionics, Mentology and Personality Psychology]. — 1996. — №4. — pp. 16–19.
3. Bukalov A.V. Potentsial lichnosti i zagadki chelovecheskikh otnosheniy [The Potential of the Personality and the Riddles of Human Relations]. — Moscow: Chernaya belka, 2009. — 592 pp.
4. Vekker L.M. Psikhika i real’nost’: edinaya teoriya psikhicheskikh protsessov [Psyche and Reality: A Unified Theory of Psychic Processes]. — Moscow: Smysl, 1998. — 685 pp.
5. Grossman I., Magnus W. Gruppy i grafy [Groups and Their Graphs]. — Moscow: Mir, 1971. — 246 pp.
6. Gulenko V.V. Gumanitarnaya sotsionika [Humanitarian Socionics]. — Moscow: Chernaya belka, 2009. — 344 pp.
7. Gulenko V.V. The information aspect // SMiPL. — 2000. — №4. — pp. 24–26.
8. Ermak V.D. Signs of the functions in the models of socionics // SMiPL. — 2012. — №5. — pp. 47–55.
9. Medvedev N.N. Psikhologicheskie ocherki. Raboty 1985–87 gg. [Psychological Sketches. Works of 1985–87]. — Moscow: Chernaya belka, 2012. — 288 pp.
10. Minaev Yu.P. 15 small groups of intertype relations that generate 23 bipolar traits for the types of information metabolism // SMiPL. — 2014. — №1. — pp. 11–18.
11. Minaev Yu.P. Arguments for the polemic on the two groups of bipolar traits // SMiPL. — 2014. — №5. — pp. 21–36.
12. Minaev Yu.P. From intertype relations to two groups of bipolar traits // SMiPL. — 2014. — №6. — pp. 5–17.
13. Minaev Yu.P. Operators of the classical intertype relations: from diagrams, tables and matrices to the canonical representation as a product of “basic” operators // SMiPL. — 2016. — №4. — pp. 40–53.
14. Minaev Yu.P. The irrational and rational languages of thinking // SMiPL. — 2018. — №5. — pp. 40–46.
15. Minaev Yu.P., Reinin G.R. Sections of the socion and bipolar traits // SMiPL. — 2018. — №3. — pp. 51–61.
16. Minaev Yu.P., Reinin G.R. Mathematical modelling of intertype relations // SMiPL. — 2018. — №6. — pp. 48–61.
17. Minaev Yu.P. Symbols of the dual dyads in “square” and “quadra” frames // Psikhologiya i sotsionika mezhlichnostnykh otnosheniy [Psychology and Socionics of Interpersonal Relations]. — 2014. — №9–10. — pp. 43–50.
18. Minaev Yu.P., Datsenko I.P., Pinda M.V. Simple rules for multiplying bipolar traits // PiSMO. — 2017. — №11–12. — pp. 25–35.
19. Reinin G. Tayny tipa. Modeli. Gruppy. Priznaki [Secrets of Type. Models. Groups. Traits]. — Moscow: Chernaya belka, 2010. — 296 pp.
20. Tsypin P.E. Signs of the functions and the quadral worldview // SMiPL. — 2006. — №2. — pp. 14–17.
21. Churyumov S.I. Ulybka Cheshirskogo Kota, ili Vozmozhnoe i nevozmozhnoe v sotsionike [The Smile of the Cheshire Cat, or the Possible and the Impossible in Socionics]. — 2nd ed., corr. and suppl. — Book 1. — Moscow: Chernaya belka, 2011. — 408 pp.
22. Eglit I.M. Signs of the psychic functions in Model A // SMiPL. — 2007. — №5. — pp. 11–28.
23. Jung C.G. Psikhologicheskie tipy [Psychological Types] / Trans. from the German. — Moscow: Universitetskaya kniga; AST, 1998. — 720 pp.

(SMiPL = Sotsionika, mentologiya i psikhologiya lichnosti; PiSMO = Psikhologiya i sotsionika mezhlichnostnykh otnosheniy.)