The Socionic Compass

Yu. P. Minaev Socionics, Mentology and Personality Psychology, No. 1–2, 2021, pp. 19–43 UDC 159.9.075 : 159.923 (Section: Mathematical Methods in Socionics)

Abstract

This article develops the idea embedded in “Jung’s Compass.” The poles of the proposed “Socionic Compass” are designated not by the names of the psychic functions singled out by C. G. Jung, but by the graphical symbols of fours of socionic types grouped by the principle of similarity of their first functions. One of the fours comprises all the intuitives by first function. On the opposite pole is placed the graphical designation of the four of sensorics by first function. The symbols of the fours of logicians and ethicists by first function designate the two remaining poles of the “Socionic Compass.” Using the proposed infographics, all the elements of the group of operators of the classical intertype relations are obtained, as well as all the elements of two 16-element groups of central dichotomies of the socion having a common 8-element subgroup. For the elements of the named groups, formalized designations are used which make it easy to perform the binary operations in these groups.

Keywords: socionics, infographics, group of operators of the classical intertype relations and its subgroups, binary operations, groups of central dichotomies of the socion, Augustinavichiute–Reinin traits, Jung–Minaev traits.

Translator’s note: operator notation as in the companion papers (1, -1, 1*, -1*, c, -c, c*, -c*, m, -m, m*, -m*, cm, -cm, cm*, -cm*). TIM pictographs and the special pair-symbols of the socionic font (superego-pair, extinguishment-pair and dual-pair glyphs) did not survive text extraction; TIMs in running text are restored as acronyms where the prose determines them, and pair-symbol listings are indicated as such. Figures are described in brackets.

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Introduction

The “Appendix” to C. G. Jung’s “Psychological Types” [23] gives the texts of four of his works on psychological typology published in various years. One of them (1931) is based on a lecture delivered at the Congress of Swiss Psychiatrists (Zurich, 1928). This work ends with the following words:

“The four functions are a kind of four points of the horizon, as arbitrary as they are necessary. Nothing prevents shifting the point of coordinates to one side or the other, or indeed giving them other names. Everything depends on how we agree, and on how expedient it is.

But I must confess one thing: I would not for anything wish to do without this compass in my psychological research expedition — and not for the obvious human reason that everyone is in love with his own ideas, but because of the objective fact that it thereby provides a system of measurement and orientation, and this in turn makes possible the emergence of a critical psychology which we have so long lacked” [23, p. 643].

In another work (1936) from the same “Appendix” it is written:

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

A little further in the text of that work 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, i.e. with extraversion and introversion, so that the functions themselves appear in an extraverted or introverted variant. This creates a structure of eight demonstrable functional types” [23, p. 659].

It is precisely these 8 types that are analyzed in detail in the tenth chapter (“General Description of the Types”) of C. G. Jung’s fundamental work “Psychological Types.” But attention should be paid to the fact that at the very end of this chapter there is a small section titled “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 also a relatively unconscious auxiliary function which is in every respect different from the nature of the principal function. From these mixtures well-known pictures arise — for instance, practical intellect paired with sensation, speculative intellect permeated with intuition, artistic intuition selecting and presenting its images by means of judgment tinged with feeling, 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 admitted the doubling of the number of types through consideration of the second (auxiliary) function. The founder of socionics, Aushra Augustinavichiute, in her work “On the Dual Nature of Man,” explaining the “twinning” of type names she proposed, wrote:

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

An example of such a “twinned” name is the logical-sensoric extravert (ЛСЭ, LSE). This is precisely that “logical type” which comes with sensorics, and it is a type with the extraverted attitude. In the cited work, a symbolic designation of this type was used in the form of an ordered pair of the symbols of the informational aspects (IAs) processed by the first (principal, base, program) and second (auxiliary, creative) functions.

This same socionic type, or type of information metabolism (TIM), can also be designated in abbreviated form with a superscript sign. The simple rule of transition to the abbreviated symbolic designations of TIMs will be discussed below. There are three more types with logic in first place: ЛИЭ (LIE), ЛСИ (LSI), ЛИИ (LII). The whole four of these TIMs can be designated by the logic-macroaspect glyph with the ”±” superscript. Introducing analogous abbreviated symbolic designations for the remaining TIMs and their corresponding fours, it is not difficult to obtain the “folded” variant of the Socionic Compass (Fig. 1), which will be reminiscent of Jung’s Compass.

Fig. 1. The “folded” variant of the Socionic Compass. [A circle whose four poles carry the four macroaspect glyphs with ”±” superscripts: intuitives-by-first-function at the top, sensorics at the bottom (the axis of irrationality), and the logicians and ethicists on the horizontal (the axis of rationality).]

Note that the same symbols, only without superscripts, are used to designate the macroaspects of the informational flow: intuition, sensorics, logic, ethics. Exactly as in Jung’s Compass, the Socionic Compass has the axes of irrationality and rationality.

The main aim of the present work is to show how to “unfold” the “folded” variant of the Socionic Compass presented in Fig. 1, extracting from it information on 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. At issue are the group of operators of the classical intertype relations (IRs) and two groups singled out, precisely from the standpoint of the classical IRs, from the enormous set of groups of central sections of the socion having the same order.

The system of formalized designations of the operators of the classical IRs

In this section it will be shown how to obtain the group of operators of the classical IRs by means of the Socionic Compass. By the classical model of IRs we shall understand the one proposed by Aushra Augustinavichiute in her work “The Theory of Intertype Relations” [1, pp. 109–181]. This model is not directly connected with any group of bipolar traits of TIMs. The classical IRs are often considered by invoking the Models A of the communicants, though in the cited work the founder of socionics used the simplified Models Yu for the TIMs. We shall now use, as models of the TIMs, their abbreviated symbolic designations. With this approach, exactly the same group of IR operators will result.

Note that the introduction of the mathematical notion of operator into the consideration of intertype relations was proposed by G. R. Reinin [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, such IRs are called asymmetric. In other words, in the case of symmetric IRs one operator “works” in the pair of TIMs, while in the case of asymmetric ones two different operators “work,” which are inverses of each other.

If the pair of communicants comprises two identical TIMs, one speaks of the identity relations, and the corresponding operator is called the identity operator. We shall designate this operator by the symbol “1.” In the group of IR operators it plays the role of the unit element. In the socionics literature this operator may be designated as “Т,” “ТО,” “ТЖ.” Obviously such designations are tied to the words used in naming the IRs. But they do not reflect the mathematical structure of the group of operators; they are not adapted to performing the binary operation in this group. From their outward appearance one cannot say how the models of TIMs are transformed under their action. We shall compare our formalized designations of the operators of the classical IRs with those used in V. V. Gulenko’s book “Humanitarian Socionics” [6]. That book adopts a system in which the designations of at least four operators of asymmetric IRs stand out visually. In our formalized system of designations, the visual images of the operators carry far more information about their mathematical properties.

Before proceeding directly to the group of operators of the classical IRs and the introduction of the formalized designations, let us give brief explanations of the rule for the use of signs in the abbreviated symbolic designations of TIMs. Since these designations are used in the socionics literature not very often, let us show, by means of the Socionic Compass, how to find from the sign in the abbreviated designation of a TIM the IA of its second function.

Look at the left-hand scheme in Fig. 2, where the Socionic Compass is presented so that the abbreviated symbolic designations of all 16 TIMs forming the socion are visible. If the sign ”+” is used as the superscript in the abbreviated symbolic designation of a TIM, then the informational aspect of the second function will be the same as that of the first function of the nearest TIM when moving clockwise along the corresponding circle. In the case of the sign ”‒,” one must move counter-clockwise.

Take as an example ESE⁺-type notation with the ”+” sign: moving along the corresponding circle clockwise, we run into the neighboring symbol, which fixes its second function; the example in the text gives ▙⁺ = EIE (the ethical symbol with ”+” unfolds to ethics-intuition). Consider a TIM with the other sign: moving counter-clockwise we obtain the unfolding of the ”‒” designation, the example being LII.

Looking ahead, let us point out that the sign in the abbreviated symbolic designation records the TIM’s membership in a pole of the bipolar trait right / left (+ / ‒ respectively). Such an indication may prove useful for those with whom clockwise rotation (of a steering wheel, say) is associated with a right turn, and counter-clockwise with a left turn. It must be noted that the connection between the shapes of the IA symbols of the first and second functions does not depend on the “color.”

Fig. 2. Explanation of the sign rule in the abbreviated symbolic designations of TIMs, and introduction, using the Socionic Compass, of the formalized designations of the operators in the cases of symmetric IRs.

[Two “unfolded” compasses, each a cross of 16 TIM symbols on four concentric circles. Left: connecting lines of three kinds mark the pairs 1* (kindred, solid), c (extinguishment, dashed), and the derived c* = 1* • c (mirage); annotation (1*)² = c² = m² = (-1)² = 1 /identity/. Right: bands mark m (mirror, blue), -m = -1 • m (conflict, yellow), the dotted line -1 (super-ego), plus the derived symmetric products -1* = -1 • 1* (business), -c = -1 • c (duality), -c* = -1 • c* (semi-duality), cm = c • m (quasi-identity), -cm = c • (-m) (activation).]

Using this same scheme, we become acquainted with the formalized designations of three more operators (in addition to the operator 1) which do not change the shape of the IA symbol of the first function. The designation 1* indicates that only the sign used as the superscript must be changed. For example, 1* applied to EIE gives ESE. In the unformalized designations of IR operators using the three-letter codes of TIMs, this formula would look like: РО (ЭИЭ) = ЭСЭ, and the corresponding phrase would be: the kindred of the ethical-intuitive extravert is the ethical-sensoric extravert. But one can also say simply: EIE and ESE are kindred. Such a replacement is possible because the operator 1* is its own inverse (is an involution). In other words, the kindred IRs are symmetric. From the standpoint of the change of the abbreviated symbolic designation of a TIM under the action of the operator 1*, this fact is completely obvious: changing the sign twice, we return to the original. Also correct in socionics is the phrase: the kindred of the kindred is the 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 the kindred types, which leads to the transition from Jung’s 8-element typology to the 16-element typology adopted in socionics.

The second operator for which a formalized designation is introduced on the same scheme is the operator of extinguishment (complete opposition, neutralization). The action of this operator on the abbreviated symbolic designation of a TIM is well visible. It changes only the color of the symbol, leaving unchanged not only the shape but also the superscript (sign). Therefore the letter “c” (from the English word colour) is used to designate this operator. If this letter is read not as “cee” but as “tse,” it may remind the Russian-speaking reader of the word “цвет” [color]. This operator too is its own inverse (is an involution). Indeed, changing the color twice, we return to the symbol of the initial TIM.

Having two operators, one can “multiply” them, introducing a binary operation on the set of operators. The attentive reader may have noticed that we already used the binary operation without special warning when multiplying the operator 1* by itself. But when we need to multiply two different operators, the order of the factors may prove important. The question arises: are the kindred of the extinguishment partner (1* • c) and the extinguishment partner of the kindred (c • 1*) one and the same?

That one should be careful with such questions is suggested by the experience of ordinary spoken language. It is clear, after all, that “the sister’s son” is not the same as “the son’s sister.” But in our concrete case the product of the operators 1* and c does not depend on the order of the factors. In both cases the operator of the mirage relations results, for which we introduce the designation c*. In such cases the operators are said to commute with each other. Therefore not only the equality c* = 1* • c shown in the figure is valid, but also the equality c* = c • 1*.

The stated commutativity of the operators 1* and c is completely obvious from the standpoint of our formalized designations. Indeed, it is clear to everyone that the operations of changing the sign and changing the color can be interchanged. The result will not change.

Note that in every four of TIMs there are 6 pairs. On the left-hand scheme (see Fig. 2) the operators of the relations are shown by the corresponding connecting lines in only three pairs. But this is quite sufficient to understand what the relations are in the remaining three pairs. Take, for example, the four of intuitives by first function. Acting with the operators of the triple {1*, c, c*} on any TIM of the indicated four, we obtain the other three TIMs. For example, 1* (ILE) = IEE, c (ILE) = ILI, c* (ILE) = IEI. Let us stress once more the fact that the operators analyzed do not change the shape of the IA symbol of the first function of the TIM on which they act.

Let us pass to the right-hand scheme in Fig. 2. It is intended for acquaintance with the formalized designations of the remaining operators which are their own inverses (are involutions). All of them change the shape of the IA symbol of the first function of the TIM on which they act. Let us begin with the operators of mirrorness (m, from the English mirror), super-ego (-1) and conflict (-m), for which special kinds of connecting lines are provided on the scheme.

From the standpoint of the effect on the abbreviated symbolic designations of TIMs, the simplest of them is the operator super-ego (-1). It changes neither the sign nor the color, and changes the shape quite unambiguously (within one, as they say, rationality pole). One may say that in the case of this operator the initial and resulting TIMs belong to one pole of the dichotomy irrationals / rationals. The designations of TIMs linked by the super-ego (-1) relation are located on the scheme symmetrically about its center.

The operator of mirrorness (m) changes sign, color, and shape. Moreover, it changes the shape depending on the sign of the initial TIM. There will be no problems with this operator once the skill of handling the abbreviated symbolic designations of TIMs has been worked up. One must learn to turn them mentally into the expanded ones (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 in which, in the EGO block consisting of the first two functions, the order of the IA symbols differs. For example, the expanded equality m(ILE-expanded) = LII-expanded; after replacing the designations by the abbreviated ones, the same equality is rewritten with the short symbols.

As is easy to guess, the conflict operator (-m) is the product of the two operators just analyzed, and the order of the factors does not matter. The super-ego of the mirror partner is the conflicter, and the mirror partner of the super-ego is also the conflicter. The commutativity of the operators super-ego (-1) and mirrorness (m) is easily explained in the language of transformations of the TIM designations. One need only note that under the action of the super-ego operator (-1) the shape of the IA symbol changes, within one rationality pole, not only of the first but also of the second function. Now it becomes completely obvious that the operation of changing the shape within one rationality pole (-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 reconstruct the other three elements of the four whose symbols lie in the Socionic Compass on one circle. For example, -1 (ILE) = SLE, m (ILE) = LII, -m (ILE) = ESI-quadrant partner. Recalling which fours of TIMs are linked by the operators of the triple {1*, c, c*}, it is not difficult to understand 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 already analyzed six {1*, c, c*, -1, m, -m}.

Suppose that, in the case where two “steps” are needed, the first is accompanied by a change of the symbol’s shape (which corresponds to the operators of the triple {-1, m, -m}), while at the second the shape of the symbol does not change (which corresponds to the operators of the triple {1*, c, c*}). Geometrically this can be pictured so: at the first “step” the movement proceeds along a circle, and at the second “step” along a radius. Of course, if at the first “step” the super-ego operator (-1) is to be used, the movement along the circle can be replaced by a “jump” along the diameter from the initial point to the point symmetric to it about the center of the Socionic Compass.

In Fig. 2 expressions are written out for five more operators of the classical IRs: 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 the triple {1*, c, c*}. In that figure only the cases are considered where the multiplied operators 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. is an operator of symmetric IRs. These assertions are fairly obvious if one takes into account what transformations of the abbreviated symbolic designations of TIMs the operators entering the products bring about.

Now consider Fig. 3. Note the operator m* (of transition to the supervisee). This operator is obtained as the product of the mutually non-commuting operators 1* and m, taken in a definite order. The supervisee is the kindred of the mirror partner: m* = 1* • m. For example, m* (ILE) = 1* [m (ILE)] = 1* [LII] = LSI-partner. Finding by the same method the supervisee of that result, repeated application of this operator did not return us to ILE. Taking into account that the resulting TIM is -1 (ILE), we can write the equality (m*)² = -1. Upon squaring (multiplication by itself) this operator gave not the identity operator (1) but the super-ego operator (-1).

Let us try changing the order of the operators in the product, i.e. finding the mirror partner of the kindred of ILE. Here is what results: m [1* (ILE)] = m [IEE] = the semi-opposite result. The designation for this operator (-m*) is introduced taking into account that the result equals -1 applied to the m*-image. Regardless of what the initial TIM is, the equality m • 1* = -1* • m holds, which has been called the fundamental commutation relation (the interchange rule) [13]. The resulting operator (-m*) is the inverse of the operator m*, which is written mathematically as (m*)⁻¹ = -m*, or as -m* • m* = m* • (-m*) = 1. In other words, -m* is the operator of transition to the supervisor (the controller). The square of this operator, like that of m*, equals the super-ego operator (-1).

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

cm* = c • m* (the operator of transition to the benefactor); -cm* = c • (-m*) (the operator of transition to the beneficiary).

One need only note that the benefactor, for example, is not the extinguishment partner of the supervisor but the extinguishment partner of the supervisee (c • m*), or the dual of the supervisor: cm* = -c • (-m*).

For the operators of the transitions to the supervisor (-m*), benefactor (cm*) and beneficiary (-cm*), other expressions are given in Fig. 3. This is connected with the wish, as in the previous cases, to show the possibility of making the required transitions in two “steps,” the first of which corresponds to an operator of the triple {-1, m, -m} (movement in the Socionic Compass along a circle), and the second to one of {1*, c, c*} (movement along a radius). At the first “step” the shape of the symbol changes, and at the second it does not.

Fig. 3. The three-element system of generators of the group of operators of the classical IRs, and the decomposition of the canonical representation of the operators into two factors convenient for work with the Socionic Compass.

[Left: the compass with connecting lines of only three kinds (1*, c, m). Right: the compass with the -1 (dotted) and -m (yellow) bands added, plus the boxed formula (-1)ᵏ • cˡ • (1*)ᵖ • m^q = [(1*)ᵖ • cˡ] • [(-1)ᵏ • m^q], k, l, p, q ∈ {0; 1}; annotations (m*)² = -1; m* = 1* • m /Р–/; -m* = 1* • (-m) /Р+/; cm* = c* • m /З+/; -cm* = c* • (-m) /З–/.]

Note the left-hand scheme in Fig. 3. In this variant of the Socionic Compass, connecting lines of only three kinds (1*, c, m) have deliberately been left. Nevertheless, along these connecting lines one can get from the symbol of every TIM to any other. This means that the triple of operators {1*, c, m} can serve as a system of generators of the 16-element noncommutative group of operators of the classical IRs. True, the expressions for the other operators in terms of the operators of this triple may turn out rather long. Let us demonstrate this with a concrete example. Suppose we need to “get” from ILE to SEI. One of the possible, not shortest, routes along the remaining connecting lines is a five-step chain. The sequence of operators effecting the required transformations gives the following expression for the duality operator (-c), which transforms ILE directly into SEI: m • 1* • m • c • 1*. One must only remember that with such a notation of a product of operators it is assumed that the transformations are performed in reverse order (from right to left), i.e. first 1*, then c, then m, and so on.

Let us demonstrate on this same example that the use of the formalized designations of the IR operators makes it possible to compute such products mentally. The operator c commutes with all operators. Consequently it can be moved to any place. All the operators entering our system of generators are involutions (their squares equal 1). Multiplication of any operator by the unit operator (1) does not change the former. 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 super-ego operator (-1), represented by the “minus,” can likewise be moved to any place — it, like c, commutes with all operators. Moreover, the associativity property obligatory for groups was taken into account, allowing us to place parentheses in the right spots, which suggest which “stretches” of the given 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 system of generators, into which the super-ego operator (-1) is included as the fourth element, then for any operator of the classical IRs one can write its canonical representation in the form (-1)ᵏ • cˡ • (1*)ᵖ • m^q, 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 four {-1, c, 1*, m} occurs at most once. If all the exponents equal zero, this is the identity operator (1). The longest path, if one uses only the operators of this four, consists of four “steps” and corresponds to the operator of transition to the beneficiary (-cm*). In this case the operator m acts at the first “step.” The order of action of the remaining operators after that no longer matters, since they commute among themselves.

In the discussion of Fig. 2 it was said that in the Socionic Compass one can get from the symbol of one TIM to the symbol of any other in at most two “steps,” using at the first “step,” for movement along a circle, the operators of the triple {-1, m, -m}, and at the second “step,” for movement along a radius, those of the triple {1*, c, c*}. For the case of the operator of transition to the beneficiary (-cm*), the first “step” will correspond to the conflict operator (-m), and at the second “step” one must perform the transformation described by the operator c* (of the mirage IRs). Indeed, c* • (-m) = -cm*.

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

(-1)ᵏ • cˡ • (1*)ᵖ • m^q = [(1*)ᵖ • cˡ] • [(-1)ᵏ • m^q].

Such a decomposition into two factors eases the mental finding of the operator that transforms the initial TIM into the resulting one. The factor [(-1)ᵏ • m^q] secures the required shape of the symbol of the resulting TIM, while the factor [(1*)ᵖ • cˡ], which corresponds to the second “step,” makes it possible to obtain the required color and sign.

Let us consider a concrete example of the task of finding an IR operator from the known initial and resulting TIMs. With which operator must one act on ESE to obtain SEE? To obtain the required shape, of the operators of the form [(-1)ᵏ • m^q] only m fits (k = 0, q = 1). It remains only to change the color. One of the operators of the form [(1*)ᵖ • cˡ] is “responsible” for this — namely c (p = 0, l = 1). Thus the sought operator is found as the product c • m = cm, i.e. what is at issue is the operator of quasi-identity.

And here is an example of finding the resulting TIM from the initial TIM and the operator. How to find the benefactor of LII? Since the operator cm* is representable as the product c* • m, two “steps” must be made: c* [m (LII)] = c* [ILE] = IEI — i.e. the benefactor of LII is IEI.

Finally, an example must be analyzed of the task of finding the initial TIM from the resulting TIM and the operator. Let us show that such a task easily reduces to the preceding kind. If the operator is an involution (i.e. the IRs are symmetric), one must simply act with the given operator on the resulting TIM. Compare the two questions:

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

In both cases the answer will be the same, since the semi-dual relations are symmetric. Therefore, in answering the first question, we can proceed just as in answering the second, taking into account the representation of the given operator as the product -c* = c* • (-1). Then we obtain: -c* (IEE) = c* [-1 (IEE)] = c* [SEE] = SEI. One may say that IEE and SEI are semi-duals.

The case with asymmetric IRs is not much more complicated. Suppose there is the question: for which TIM is IEE the supervisor? One need simply reformulate it thus: which TIM is the supervisee of IEE? And then we proceed by the old algorithm:

m* (IEE) = 1* [m (IEE)] = 1* [EII] = ESI.

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

Division of the socion by means of proper subgroups of the group of operators of the classical IRs

The formalized designations of the operators of the classical intertype relations, like the symbolic designations of the TIMs, and the Socionic Compass itself, are useful elements of infographics, which makes it possible not only to present socionic information in compact and visual form, but also to obtain consequences in a simpler and more comprehensible way. How to find out quickly whether two given operators of the classical IRs commute with each other? It turns out this can easily be done from the outward appearance of their formalized designations.

Looking attentively at the proposed operator designations, 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 designations exactly in half, and the corresponding dichotomies are independent and pairwise orthogonal. By indicating a concrete set of poles of these bipolar traits, an operator of the classical IRs is specified unambiguously.

Note that every formalized designation of an operator of asymmetric IRs necessarily contains both the letter m and the “asterisk” (). And this sets these operators apart from the rest. On the other hand, there are four operators whose designations contain neither the letter m nor the “asterisk” (). In the four {1, -1, c, -c} are gathered all those operators which commute not only among themselves but with all the rest. This four is itself a group in the mathematical sense of the word and, correspondingly, a subgroup of the whole 16-element noncommutative group of operators of the classical IRs. The subgroup gathering all elements that commute with the rest is called the center of the group. The whole group can be decomposed into cosets with respect to the center. In the given case the following decomposition results:

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

As we see, with this decomposition all the operators of asymmetric IRs automatically gathered in the last coset. In the coset {1*, -1*, c*, -c*} the designations of all 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 each coset two of the four operators contain in their designations the letter c, and two the “minus” (-). This is precisely where the independence and orthogonality of the central sections of the considered set of operator designations by the indicated bipolar traits is reflected.

If one takes any element of the last three cosets and multiplies it by the elements of the center, the whole corresponding coset results. That is in fact how the decomposition of the whole group into cosets with respect to its center is carried out. Operators not belonging to the center commute only with operators of the center and of their own coset. When two operators of one and the same coset are multiplied, the product belongs to the center. If one multiplies two operators from different cosets, neither being the center, the product belongs to the third coset that is not the center. Since an operator’s membership in a given coset with respect to the center is well visible from the outward appearance of its formalized designation, the question of whether two concrete operators commute or not should cause no difficulty.

Let us point out one more circumstance worth remembering. If two operators of the classical IRs do not commute with each other, their two possible products differ only in sign (interchange of the factors is equivalent to multiplication by the super-ego operator (-1)). In other words, for any two mutually non-commuting operators an interchange rule analogous to the fundamental commutation relation holds. 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*, and so on.

Using the formalized designations of the operators of the classical IRs, it is not difficult to find all the proper subgroups of their 16-element noncommutative group. At issue are the 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). With such divisions, in every part one and the same set of IR operators will be realized, allowing one to find, from one TIM of a given part, all the others belonging to that part of the socion. The set of operators is obtained from the corresponding subgroup by excluding the unit element, i.e. the identity operator (1).

The most obvious case is that of the subgroups of order 2. Each such subgroup comprises the unit operator (1) and one more operator of symmetric IRs. And of such operators 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 these pairs by one of the listed operators. In other words, in this way the socion can be divided either into 8 super-ego pairs (-1), or into 8 extinguishment pairs (c), or into 8 dual pairs (-c), or into 8 pairs of kindred (1*), and so on down the list.

The subgroups of order 8 (and the corresponding dichotomies of the socion) number 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 in this list contain within themselves the center of the whole group plus one whole coset with respect to the center. The third subgroup differs from the first two in that all 4 operators of asymmetric IRs entered it. The first two subgroups consist exclusively of operators of symmetric IRs. Each of the subgroups numbered 4–7 in this list comprises two elements from each coset with respect to the center, differing from each other in sign. Obviously, no other 8-element subgroups of the group of operators of the classical IRs exist. And this means that there exist only 7 central dichotomies of the socion such that each half of the socion can be reconstructed from its single representative by means of the operators of the corresponding subgroup of order 8.

Let us consider for now only one illustrating example. Take ILE and act on it with the operators of the first subgroup in our list. We obtain a sequence of eight TIMs — all the irrationals. If the same procedure is carried out taking, say, SLE as the initial TIM, the composition of the eight TIMs does not change: again all the irrationals are written out. Had the initial TIM been rational, the result would have been the eight of rationals. Acting with the same sequence of operators on, say, EII, we obtain the eight rationals. Thus, to the first subgroup of order 8 in our list one can put into one-to-one correspondence 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 now, with the help of suitable infographics, let us make visual the differences among the seven subgroups of order 8 under discussion. For this we use the so-called Cayley graphs [5], which for our case are presented in Fig. 4.

Fig. 4. Cayley graphs for the three varieties of subgroups of order 8 of the group of operators of the classical intertype relations. [Left: the cube-like graph of C2×C2×C2 with three edge-colors; middle: the graph of C4×C2 with directed edges for the C4 generator; right: the octagonal graph of D8.]

The first two 8-element subgroups in our list can be represented as the direct product of three of their cyclic subgroups of order 2. In the first case the triple can be {1, -1}, {1, c}, {1, 1*}, and in the second {1, -1}, {1, c}, {1, m}. Correspondingly, the minimal system of generators will consist 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 graph in Fig. 4). The operators entering the system of generators are additionally represented by the edges of the Cayley graph, which differ from one another in some way (for example, by color). At every vertex, edges of all three varieties meet. If one of the vertices of the Cayley graph is put into correspondence with the unit element of the group (1), and each variety of edge with its own element of the system of generators, then the correspondence between the remaining vertices of the graph and the elements of the group is thereby fixed as well.

The third subgroup of order 8 is unique of its kind. To it corresponds the middle graph in Fig. 4. It is representable in terms 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 the form C4 in this case can be generated by the single operator m*. Therefore the system of generators of the whole 8-element group can be reduced to two elements ({m*, c}). Since the operator m* is not an involution (is not its own inverse), on the Cayley graph directed edges (with an arrow) correspond to it.

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

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

Let us pass now to the consideration of the 4-element subgroups. Subgroups of this order number 15 in all. Only two of them are cyclic subgroups of the fourth order (C4), whose system of generators consists of just one element:

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

The remaining 13 subgroups of order 4 contain no operators of asymmetric IRs and can be represented as the direct product C2×C2. Fig. 5 gives the corresponding Cayley graphs for the two indicated varieties of subgroups.

Fig. 5. Cayley graphs for the two varieties of subgroups of order 4 of the group of operators of the classical intertype relations. [Left: the directed square of C4; right: the two-colored square of C2×C2.]

Although the 13 subgroups of order 4 are isomorphic to one another, it is useful to distinguish them by whether operators of the triple {-1, c, -c} enter their composition. All three operators enter the center of the group ({1, -1, c, -c}). Only one operator of this triple enters the following subgroups of order 4:

{1, -1, 1*, -1*}, {1, -1, c*, -c*}, {1, -1, m, -m}, {1, -1, cm, -cm} — super-ego (-1); {1, c, 1*, c*}, {1, c, -1*, -c*}, {1, c, m, cm}, {1, c, -m, -cm} — extinguishment (c); {1, -c, 1*, -c*}, {1, -c, -1*, c*}, {1, -c, m, -cm}, {1, -c, -m, cm} — duality (-c).

Let us demonstrate how a tetrachotomy of the socion is carried out by means of subgroups of the fourth order of the group of operators of the classical IRs, with the example of the division into quadras — i.e. into fours of TIMs in which the relations of duality (-c), mirrorness (m) and activation (-cm) are realized. Clearly what is at issue is the subgroup {1, -c, m, -cm}. Taking an arbitrary TIM and acting on it with the operators of the indicated subgroup, we obtain one of the quadras. Then one must take a TIM not obtained in this way and act on it with the operators of the same four. After the third cycle, the TIMs of the last quadra remain. Of course, with such a procedure the order of the resulting quadras will be random, but the division of the socion will in any case be into quadras and not into any other fours of TIMs. The quadras are designated by Greek letters in this order:

{ILE, SEI, LII, ESE} — α; {LSI, EIE, SLE, IEI} — β; {SEE, ILI, ESI, LIE} — γ; {EII, LSE, IEE, SLI} — δ.

In an analogous manner the socion can be divided into 4 parts by means of the remaining 14 subgroups of order 4. The fours of TIMs obtained in such divisions G. R. Reinin proposed to call homogeneous [19, p. 172]. What was meant is that in such a four all the TIMs entering it are “in an identical intertype situation.” Note that among the homogeneous fours (from the standpoint of the classical IRs) also fall those 8 fours to which correspond the two subgroups of the form C4, consisting of two mutually inverse operators of asymmetric IRs (m* and -m*, or cm* and -cm*), the super-ego operator (-1), and the identity operator (1) as the unit element of the group. Of course, the identity operator (1) is not realized in fours consisting of distinct TIMs. Therefore it was not included in the “formula” of the four [16]. Since to each of the 15 “formulas” obtained by discarding the unit element from the subgroups of order 4 there correspond 4 fours of TIMs, there exist 60 homogeneous fours in all.

The division of the fours of TIMs into homogeneous and inhomogeneous arose in connection with the fact that G. R. Reinin divided the socion into 4 parts in a fundamentally different way, using pairs of bipolar traits from the group he constructed, whose system of generators was the four: extraverts / introverts, irrationals / rationals, intuitives / sensorics, logicians / ethicists. This four is usually called the “Jung basis,” although the last two elements in it fundamentally could not have appeared in Jung’s 8-element typology, in which the kindred types were not distinguished [15].

When the socion is divided by means of pairs of ARTs (Augustinavichiute–Reinin traits), only in 15 of the 35 possible cases do the fours of TIMs come out homogeneous. But of these 15, only in 11 cases was there a common “formula” for all four fours. Four tetrachotomies of the socion organized by means of pairs of ARTs turned out “two-formula”: for the fours consisting of irrationals the “formula” differed from the one corresponding to the fours of rationals. By means of the ARTs it is impossible to organize “one-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 fours of TIMs can be obtained by means of two suitable central dichotomies. This means that besides the ART group there must exist at least one more group of bipolar traits which is interesting from the standpoint of the classical intertype relations. Such a group was constructed and received the name of the JMT group (Jung–Minaev traits) [11]. Its construction was connected with the wish to verbalize precisely those dichotomies of the socion whose pairs allow one to organize the tetrachotomies of the socion corresponding to the 4-element subgroups just listed of the 16-element noncommutative group of operators of the classical IRs.

Now we shall show 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. Let us begin with the JMT group, where, unlike in the much better known ART group, one can find a triple of bipolar traits that could serve as a basis for Jung’s 8-element typology.

The Socionic Compass as the basis for constructing two 16-element groups of bipolar traits of TIMs with a common 8-element subgroup

In constructing the JMT group we shall need the convention, recorded in Table 1, concerning the names of the bipolar traits of the informational aspects. With this table as an example we shall also demonstrate the principle of introducing a binary operation on the set of bipolar traits of the IAs.

Table 1. The group of bipolar traits of the informational aspects used.

Rows (the eight IAs form the columns, with black intuition as the reference column of all ”+”):

#	Bipolar trait
0	IA / not-IA (the existence trait)
1	irrational / rational
2	detached / involved
3	internal / external
4	static / dynamic
5	extraverted / introverted
6	α / γ values
7	δ / β values

[The body of the table assigns ”+”/”‒” to each of the eight IA symbols per row; the black-intuition column is all ”+“. Traits 1, 2 and 4 (bold) are taken as the “basis” (system of generators).]

The signs ”+” and ”‒” in the table designate the membership of an IA in the first and second pole of the bipolar trait respectively. That the reference column contains only “pluses” means that this IA is taken in the considered set as the “point of reckoning.” Row “0,” filled entirely with “pluses,” is responsible for the so-called identity section. The corresponding trait is called the existence trait. In the group it plays the role of the unit (neutral) element. Traits “1,” “2” and “4,” set in bold, are adopted as the “basis” (system of generators).

Having given, by ordered sets of “pluses” and “minuses,” sections “1” and “2” of the set of IAs, we obtain section “3” by a simple rule: if in one and the same column sections “1” and “2” have identical signs, then section “3” has ”+” in that column, and if different, ”‒” must 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 informational macroaspects, the only possible group of central sections would be given by rows “0”–“3” of our Table 1. Of course, the number of columns in the “body” of the table would be half as large. For in rows “0”–“3” the signs in the columns of the two IAs of one macroaspect coincide anyway. For the remaining pairs of IAs entering one macroaspect the situation is entirely analogous. One more change that would have to be made concerns the neutral element: the caption of the row “IA / not-IA” would have to be replaced by “macroaspects / not-macroaspects.”

Dividing each macroaspect into two IAs by the trait static / dynamic, which goes in Table 1 under number 4, we automatically obtain, by means of the binary operation, three more sections (“5”–“7”). For this, row number 4 must be “multiplied” by rows “1”–“3” respectively. Consider as an example the equality “3” ⊗ “4” = “7” in expanded form:

(internal / external) ⊗ (static / dynamic) = (δ-values / β-values).

Let us explain the introduced binary operation on this concrete example somewhat differently. To the first pole of the product belong those elements which simultaneously belong to the first poles of the factors, and also those which simultaneously belong to the second poles of the factors. The informational aspects which are simultaneously internal and static number just two; those which are simultaneously external and dynamic likewise number two. Hence to the first pole of the product belongs a four of IAs (the δ-values). Correspondingly, to the second pole belongs the four of the remaining IAs (the β-values).

Of course, such words for fixing the poles of this section of the set of IAs could be applied only after the fours of TIMs that were named quadras had received their designations. Let us note in passing that the verbalization of a concrete section of a set is a separate task, which is not solved by formal-mathematical methods. If the poles of the dichotomies of a set are given by enumeration of elements, then the construction of the whole group from a known system of generators is a purely technical task. Whereas disputes about the choice of the most felicitous terms for verbalizing a concrete section can go on very long. For example, for this same section V. V. Gulenko proposed the variant “inertial / motive” [7]. For fixing certain other central sections of the set of IAs he likewise uses words differing from those in Table 1. The system of terms he proposed for verbalizing the elements of the group under discussion could, without particular changes, be applied to Jung’s 8-element typology, in which the psychological type is in effect given by the informational aspect of the first function. But the terms figuring in the last two rows of Table 1 would not suit that purpose. This is connected with the fact that in Jung’s 8-element typology the very notion of the quadra fundamentally could not arise.

As was already said, the 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 function. The point is that the IAs 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 designations of TIMs using the superscripts ”+” (for the right) and ”‒” (for the left). Let us note in passing that in article [14] an attempt was made to explain the existence of such a rule from the standpoint of L. M. Vekker’s Unified Theory of Psychic Processes [4].

Now let us imagine that we have decided to build a table of bipolar traits of TIMs, making use of Table 1 for the IAs. We double the number of columns in the “body” of the table by a very simple principle, whereby in place of, say, the black-intuition column there appear the columns of ILE and IEE. The filling of these columns with “pluses” and “minuses” up to and including row “7” will be identical for the kindred types, since these rows would be tied to the IA of the first function only. Of course, the verbalization of the sections will have to be corrected slightly. But at present what is at issue is the formal-mathematical method of obtaining the central sections of the socion entering one of the possible 16-element groups.

In row “8” one would have to put “pluses” for the right TIMs and “minuses” for the left. This row will correspond to the fourth element of the system of generators 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 result into rows “9”–“15” respectively. Thus the “body” of the whole table for one of the possible 16-element groups of central dichotomies of the socion is obtained. Having filled in the “body” of the table, one will have to think about the verbalization of the resulting sections of the socion (about how to caption the rows).

Guided by the procedure set forth, one can obtain that group of bipolar traits of TIMs which is now called the JMT group. With the verbalization of the elements of this group no great problems arose. In the JMT group, besides the 8 elements entering the ART group well known to socionists, there turned out to be 8 more, for which terms were proposed tied to the traits of the IAs processed by the first or second functions of the TIMs [12].

Let us show how, in constructing this same group of bipolar traits, the infographics based on the Socionic Compass can be used. To simplify the schemes we shall use conventional designations for kindred pairs and extinguishment pairs. For example, the intuition glyph with ”±” will designate a kindred pair, and the two-tone intuition glyph an extinguishment pair. The two necessary schemes are presented in Fig. 6.

Fig. 6. Use of the Socionic Compass for introducing one of the possible systems of generators of the JMT group.

[Left: the compass on kindred pairs with two diagonal cutting lines B₀ and B₁ and the circle A₀; legend: B₀ = detached/involved-1; B₁ = internal/external-1; A₀ = statics/dynamics. Right: the same cross on extinguishment pairs with the cutting lines B₀, B₁ and the circle I₃ = right/left.]

Without dwelling for now on the principle of introducing the letter-digit designations for the bipolar traits, let us make a few preliminary remarks. Recall that in verbalizing the sections of any set by bipolar traits one must agree on how the order of the poles will be chosen. This is important for introducing the binary operation on the set of traits. Considering the set of IAs, we took black intuition as the “point of reckoning.” In Table 1 the corresponding column is filled entirely with “pluses.” In each row of that table, “pluses” were received by those IAs which belong to the same pole of the corresponding bipolar trait as black intuition. We did not give the tables for the macroaspects and the TIMs, but in the descriptions of how they can be built it was assumed that in them the intuition macroaspect and ILE respectively are taken as the “points of reckoning.”

In considering the 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 ILE. If the abbreviated symbolic designations of TIMs are not indicated explicitly on the schemes (see Fig. 6), one must orient oneself by the symbolic designations of the pairs or fours into which they enter. This convention makes it possible, on the schemes, to make compact captions with callouts for the lines symbolizing the sections of the socion.

Let us see, for example, how the characteristics of ESE are presented in Fig. 6. It is not difficult to discover that this TIM belongs to the first pole of the trait B₁ (like ILE), and to the second pole of the traits A₀, B₀ and I₃ (not the same as ILE). Hence ESE must 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 IAs indicated in Table 1, and know how the TIMs of one kindred pair are separated by the trait right / left, we can satisfy ourselves of the correctness of the result obtained by discovering all these characteristics directly in the abbreviated symbolic designation of the TIM.

The left-hand scheme in Fig. 6 well demonstrates 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-hand scheme indicates that the triple {B₀, B₁, I₃} is a “basis” of the 8-element typology of extinguishment pairs. A concrete TIM is fully specified by indicating these two kinds of pairs into which it enters. Consequently, the four of bipolar traits {B₀, B₁, A₀, I₃} can serve as a “basis” for the typology of socionic types. Knowledge of which poles of these four traits a given TIM belongs to makes it possible, in principle, to find out to which pole of any (not necessarily verbalized!) dichotomy of the socion it belongs.

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

Having settled on one of the variants of the system of generators 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, allotting rows “1,” “2,” “4” and “8” to the “basis” sections {B₀, B₁, A₀, I₃} (and not necessarily in that order). The remaining rows are filled in automatically by the rule indicated earlier, which realizes the binary operation adopted on the set of bipolar traits.

Note that, in introducing the indicated binary operation, G. R. Reinin explained it using a scheme similar to those given in Fig. 6. 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 schemes presented in Fig. 6, by contrast, the circle has the meaning of one more central section, which together with the sections represented by straight lines enters the “basis” of the 8-element typology of pairs of TIMs (in the left scheme — kindred, in the right — extinguishment). As for the identity section, one may say that in our schemes it is represented by the outline of the “cross” as a constituent part of the Socionic Compass.

How to 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 scheme in Fig. 6. As we already said, to the first pole of the product belong those elements which simultaneously belong to the first poles of the factors, and also those which simultaneously belong to the second poles of the factors. From the scheme it is not difficult to determine that simultaneously to the first poles of the factors belong the TIMs entering the two intuitive kindred pairs, and to the second — those entering the two sensoric kindred pairs. 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).

Let us find, using this same scheme, the “product” of the sections B₀ and A₀. To the first poles of these sections simultaneously belong the intuitive and one logical kindred pair, and to the second the sensoric and the other. If one attends to the IA symbols of the first functions of the TIMs entering these kindred pairs, one can recognize in the resulting four the α-values (see Table 1). Hence the section corresponding to the product can be verbalized as α / γ – 1, where the “1” must be read as “values by the first function.” Here is the notation in abbreviated form: B₀ ⊗ A₀ = AB₀.

Let us consider a case more difficult for verbalization, using the right-hand scheme, on which the abbreviated symbolic designations 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 preceding cases, we find that to the first pole of the product must be assigned the TIMs entering a certain four of extinguishment pairs. How to verbalize the corresponding section? Let us write out all the TIMs entering the indicated four of extinguishment pairs in expanded symbolic form. Now let us attend to the IAs processed by the second functions of the TIMs of the resulting eight. They belong to the first pole of the trait detached / involved, which was used for the set of IAs (see Table 1). This gives grounds to propose for the resulting section of the set of TIMs the name detached / involved – 2, where the “2” must be read as “by the second function.” Here is the notation in abbreviated form: B₁ ⊗ I₃ = B₂.

Let us turn once more to the right-hand scheme and try to find the section I₂, designating 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 of symmetry of the “cross.” Directly in Fig. 6 one can read that I₃ = right / left, and see that on the right-hand scheme this bipolar trait is represented by a dividing line in the form of a circle. The general rule of multiplication of bipolar traits tells us that to the first pole of I₂ belong the right irrationals (the first poles of I₃ and I₁) and the left rationals (the second poles of I₃ and I₁). The TIMs of interest to us enter a certain four of extinguishment pairs. By what suitable word to unite them?

If the found extinguishment pairs are separated into individual TIMs and the latter distributed over two fours — {ILE, SEI, LII, ESE} and {SEE, ILI, ESI, LIE} — then in these fours one can recognize the quadras α and γ. In socionics these quadras are called democratic, in contrast to the aristocratic ones (β and δ). Hence I₂ = democrats / aristocrats.

The letter-digit designations for the elements of the ART and JMT groups were introduced specially so that the binary operation in these groups could be performed quite simply, without resorting to cumbersome tables or schemes. Since these groups have a common 8-element subgroup, it made sense to choose “bases” (systems of generators) for them differing in only one element. The system of generators for the common subgroup in these designations took the form {I₁, I₂, A₀}, abbreviating the expanded record {irrationals / rationals, democrats / aristocrats, statics / dynamics}. Supplementing this triple with the trait B₀ (detached / involved-1), we obtain the “basis” of the JMT group. In the ART group, as the fourth element of the “basis,” the bipolar trait C₀ (judicious / decisive) was taken [18].

It proved possible to introduce the letter-digit designations of the bipolar traits in such a way that, in performing the binary operation, the digit and letter parts transform independently of each other. Two isomorphic groups of order 4, representable as C2×C2, resulted: {0, 1, 2, 3} and {I, A, B, AB}. The unit (neutral) elements in them are “0” and “I” respectively. Recall that we have already encountered such groups. The right graph in Fig. 5 is their Cayley graph. Such are 13 of the 15 subgroups of the 4th order of the group of operators of the classical IRs. Isomorphic to them is also the group of central bipolar traits of the informational macroaspects. Let us use the examples of JMT products we have analyzed in detail to demonstrate the independence of the performance of the binary operation in the digit and letter parts of their designations:

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 ART group, its letter part of the formalized designations is taken from the four {I, A, C, AC}. Let us show how one of the possible “bases” for this group is obtained using the Socionic Compass. In this case the intuitive, sensoric, ethical and logical poles will be presented differently (Fig. 7).

Fig. 7. Use of the Socionic Compass for introducing one of the possible systems of generators of the ART group.

[Two compasses on individual TIM symbols with cutting lines. Legend: AC₀ = intuitives/sensorics; AC₂ = logicians/ethicists; A₀ = statics/dynamics; A₁ = extraverts/introverts. Left scheme carries the circle A₀, right the circle A₁.]

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 this group, in only one element. Instead of I₁ (irrationals / rationals), here A₀ (statics / dynamics) is used. But I₁ is easily obtained as the product A₀ ⊗ A₁. In Fig. 7 it is also not difficult to see the division of the socion by the trait I₂ (democrats / aristocrats), since it is obtained as the product AC₀ ⊗ AC₂.

Hence to the democrats belong the intuitives who are at the same time logicians (the Club of researchers [sayentists]), and also the sensorics who are at the same time ethicists (the Club of socials). This democratic half of the socion we have already written down as four extinguishment pairs.

To the aristocratic half of the socion, correspondingly, belong the intuitives who are at the same time ethicists (the Club of humanitarians), and also the sensorics who are at the same time logicians (the Club of managers). As four extinguishment pairs this half of the socion is written out analogously.

Thus, by means of two orthogonal dichotomies (AC₀ and AC₂) one can obtain the tetrachotomy of the socion into the fours 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 IRs. Recall that we have already spoken of the quadras as the fours of TIMs obtained in the tetrachotomy of the socion by means of the subgroup {1, -c, m, -cm}. Two quadras (α and γ) were named democratic, and the other two (β and δ) aristocratic. If any club can be represented as two extinguishment pairs (c), then any quadra is representable as two dual pairs (-c):

two dual pairs each — α; β; γ; δ.

Let us note in passing that such designations for the dual pairs first appeared in article [17]. Besides the designations of the dual pairs and the extinguishment pairs, we shall also need designations for the super-ego pairs. Such designations were proposed in private correspondence by Andrew Joynton. Now, thanks to the efforts of Olga Bogdanovna Karpenko, these symbols already enter the socionic font, which is convenient to use in writing works on socionics.

If I₂ = {α, γ} / {β, δ}, then what bipolar traits correspond to the central dichotomies {α, δ} / {β, γ} and {α, β} / {γ, δ}? The answer to this question can be found with the help of the left-hand scheme in Fig. 7, where there are dividing lines corresponding to the traits of the triple {AC₀, AC₂, A₀}. It will therefore not be difficult 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 more sections of the socion giving I₂ in their product can be found with the help of the right-hand scheme in Fig. 7, where there are 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. 7 the symbols of the irrationals lie, if not directly on the vertical axis of symmetry of the “cross,” then near it, and the symbols of the rationals near the horizontal one, it is not difficult to find two more sections:

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

For finding the section A₀ ⊗ I₃ = A₃ (positivists / negativists) one must use the left-hand scheme in Fig. 7, 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 abbreviated symbolic designations of TIMs. The right-hand scheme in Fig. 7, in which the circle separates the extraverts from the introverts, will help find the section corresponding to the product A₁ ⊗ I₃ = A₂ (questims / declatims).

All the central sections of the socion verbalized in the ART and JMT groups can be represented as sections of 8-element sets of pairs of TIMs. We shall consider three varieties of pairs: super-ego (-1), extinguishment (c) and dual (-c). It is precisely of such pairs that the fours of TIMs homogeneous from the standpoint of the classical IRs consist.

Let us begin with the common elements of the ART and JMT groups. The letter part of the formalized designations in this case is either “I” or “A.” In the first case any of the three named varieties of pairs of TIMs can be used to demonstrate the sections. The existence trait, responsible for the identity section and playing the role of the unit (neutral) element, is designated I₀. To the first pole of this trait belong all 16 TIMs (the whole socion). In the remaining cases, to each pole of the bipolar trait belong 8 TIMs, or 4 pairs. Using the symbols for each of the three indicated varieties of pairs of TIMs, we obtain the following result: I₁ (irrationals / rationals), I₂ (democrats / aristocrats) and I₃ (right / left) — each expressible as 4-vs-4 lists of super-ego pairs, of extinguishment pairs, and of dual pairs alike. [The original writes each pole as four pair-glyphs in each of the three pair systems.]

The second four of traits entering the common 8-element subgroup (with “A” as the letter part of the formalized designations) cannot be represented using extinguishment (c) or dual (-c) pairs, but can using super-ego (-1) pairs:

A₀ (statics / dynamics); A₁ (extraverts / introverts); A₂ (questims / declatims); A₃ (positivists / negativists) — [each pole written in the original as four super-ego-pair glyphs].

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

B₀ (detached / involved-1); B₁ (internal / external-1); B₂ (detached / involved-2); B₃ (external / internal-2) — [each pole written as four extinguishment-pair glyphs].

While the first two traits of this four are well reflected in the abbreviated symbolic designations of TIMs, to see the last two traits (B₂ and B₃) behind such designations one must learn to reconstruct mentally, quickly, from the IA symbol of the first function and the sign used as the superscript, the IA of the second function. We have already spoken of how this can be done by picturing the Socionic Compass as it is depicted in Figures 2 and 3. But it is also useful to remember the connection of the signs with the professional clubs. For example, intuition with a “plus” is the one the researchers have; hence in second place in this case will be logic, which is detached and external. And this does not depend on whether ILE or ILI is at issue.

The “AB”-traits are expressed by means of dual (-c) pairs, but super-ego (-1) or extinguishment (c) pairs will not do in this case:

AB₀ (α / γ-1); AB₁ (δ / β-1); AB₂ (α / γ-2); AB₃ (β / δ-2) — [each pole written as four dual-pair glyphs].

In this four, to see the traits AB₂ and AB₃ in the abbreviated symbolic designations of TIMs, one must again be able to “compute” quickly in one’s head the IA of the second function. Besides the methods already considered, it is useful for this to remember that in the democratic quadras the right ones are the irrationals, and in the aristocratic ones the rationals. This is well seen in Table 2, where the right dual pairs stand in first places in the quadras.

Table 2. Distribution of the dual pairs over the quadras with account taken of membership in the poles of the bipolar trait right / left. [Four rows α, β, γ, δ; each row lists its right dual pair first, then its left.]

It should be said that this order of dual pairs in the quadras is used most often in socionic publications. But there are also works whose authors insist on other variants (see, e.g., [9], [21]). If one accepts the order of dual pairs in the quadras as in Table 2, and represents the socion as a closed structure in which after the δ-quadra the α-quadra comes again, then a simple rule results: in the right TIMs, second place is occupied by an IA belonging to the values not only of their own quadra but also of the following one, while in the left — to the values of the preceding one.

Let us note once more that the names for the last 8 traits (of the fours “B” and “AB”), entering only the JMT group and not the ARTs, were proposed with direct reference to the traits of the IAs processed by the first or second functions of the TIMs. Their poles are easy to name if the TIM in question is known. But filling them with psychological meaning that would help in determining the socionic type of a concrete person is a separate, far from simple task. For now let us merely note the order of the poles in the pairs of bipolar traits {B₁, B₃} and {AB₁, AB₃}. So as not to confuse them, one must always remember that ILE is adopted as the “point of reckoning.”

Now consider the bipolar traits which enter the ART group but not the JMT group. The “C”-traits of the ART group are likewise expressed by means of dual (-c) pairs (like the “AB”-traits of the JMT group):

C₀ (judicious / decisive); C₁ (carefree / farsighted); C₂ (merry / serious); C₃ (yielding / obstinate) — [each pole written as four dual-pair glyphs].

The “AC”-traits are expressed by means of extinguishment pairs (c):

AC₀ (intuitives / sensorics); AC₁ (tacticians / strategists); AC₂ (logicians / ethicists); AC₃ (constructivists / emotivists) — [each pole written as four extinguishment-pair glyphs].

So, after the explanations of how, by means of the various representations of the Socionic Compass given in Figures 6 and 7, one can obtain all the elements of the two 16-element groups of bipolar traits of TIMs with a common 8-element subgroup, we have written out explicitly all the corresponding sections of the socion. True, in the enumeration the symbols used were not those of individual TIMs but of their pairs. And the pairs themselves, depending on the bipolar trait under consideration, were formed by means of one operator of symmetric IRs from the triple {super-ego (-1), extinguishment (c) and dual (-c)}.

Note that by means of this triple the socion is divided into four parts exactly as by using any pair from the triple of mutually dependent traits {I₁ (irrationals / rationals), I₂ (democrats / aristocrats) and I₃ (right / left)}. If the indicated triple of operators is supplemented by the identity operator (1), the center of the 16-element noncommutative group of operators of the classical IRs results (i.e. the subgroup gathering all operators commuting with the elements of the whole group). If the indicated triple of bipolar traits is supplemented by the existence trait (I₀), a group of order 4 results which enters both the ART group and the JMT group.

Since the groups of bipolar traits are commutative, any 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 decompositions of the JMT and ART groups with respect to the subgroup {I₀, I₁, I₂, I₃}, using the formalized designations, have the form:

{I₀, I₁, I₂, I₃}, {A₀, A₁, A₂, A₃}, {B₀, B₁, B₂, B₃}, {AB₀, AB₁, AB₂, AB₃} (the JMT group); {I₀, I₁, I₂, I₃}, {A₀, A₁, A₂, A₃}, {C₀, C₁, C₂, C₃}, {AC₀, AC₁, AC₂, AC₃} (the ART group).

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 ART groups with respect to the subgroup I. As for the lower digit indices used by us in the formalized designations of the bipolar traits, they formed the group {0, 1, 2, 3} precisely because the four of traits {I₀, I₁, I₂, I₃} is a group (in the mathematical sense of the word).

The 16-element ART 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} = {I₀, I₁, I₂, I₃}×{I₀, A₀, B₀, AB₀}. The latter representation as the direct product of the two indicated 4-element subgroups confirms the rule of independent multiplication of the letter and digit parts of the proposed formalized designations of the bipolar traits in the JMT group. In the ART group the situation is analogous.

The Socionic Compass and the Models A of the TIMs

In the preceding sections of this work the TIMs were represented by their abbreviated symbolic designations. This was quite sufficient to build the Socionic Compass, at whose poles were placed the symbols of the TIMs united by the macroaspect to which the IAs 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 IRs 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 with the processing of which IAs not only the first two functions but the rest as well are occupied. Let us see how the Socionic Compass is connected with the Models A of the TIMs.

The signs in the abbreviated symbolic designations of TIMs have so far been associated by us only with the bipolar trait right / left, used for distinguishing the kindred in the transition from Jung’s 8-element typology to the 16-element socionic typology. But signs can be ascribed to individual IAs depending on which other IA’s symbol stands in the same horizontal block of the Model A (EGO, SUPER-EGO, SUPER-ID, ID) of the concrete 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.

Under the approach to sign placement oriented by the combination of IAs in the horizontal blocks of the Models A, a simple rule operates: with the democrats the sign ”+” is received by the irrational aspects, and with the aristocrats by the rational ones. If one now returns to the variant of the Socionic Compass presented in Figures 2 and 3, one can see in it the Models A of all 16 TIMs.

The models of the democrats will be better visible if only the symbols of the irrational IAs with the sign ”+” and the rational ones with the sign ”‒” are retained. Fig. 8 shows, with the example of ILE, how the numbers of the socionic functions must 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 designation of the given type is located — which must now be perceived as the designation of the IA of the first function with account taken of the aspect neighboring it in the EGO block. The direction in which the numbers increase from “1” to “4” is shown by an arrow (clockwise). For the vital ring of ILE (functions 5–8) another “democratic” circle of the Socionic Compass is responsible. Attention must be paid to where the cell of the fifth function is located relative to the cell of the third function, whose IA symbol differs only in color. The increase of the numbers from “5” to “8” is given by the same arrows.

Fig. 8. Use of the Socionic Compass as the Model A of ILE as a representative of the right democrats (for every TIM the numbering of the functions begins from the corresponding cell). [The cross with cell 1 at the top (black intuition), clockwise arrows, and the inset Model A of ILE for comparison.]

For the left democrats the direction of the arrows must be replaced by the opposite. As an example, ESE is taken in Fig. 9. As we see, the position of the IA symbols has not changed; the count of the numbers must simply be begun from a different cell. In the given case this cell turned out to lie on the circle of smaller radius. This is connected with the fact that initially, in the Socionic Compass (see Fig. 2), we allotted the two outer circles to the static TIMs and the two inner ones to the dynamic. When we try to see in the Socionic Compass the Model A of a concrete TIM, only two circles remain in the field of our attention. The one with the larger radius corresponds to the static ring in the Model A.

The bipolar trait of TIMs statics / dynamics is determined by the mental ring. Since ESE is a dynamic, its mental ring corresponds to the circle of smaller radius. But this does not fundamentally affect the placement of the function numbers. The transition from one circle to the other (at the passage from the fourth function to the fifth) always occurs identically. One must orient oneself by the position of the cell of the fifth function relative to the cell of the third function.

Fig. 9. Use of the Socionic Compass as the Model A of ESE as a representative of the left democrats. [Same layout, counter-clockwise arrows, start cell on the inner circle.]

What changes in the case of the aristocratic TIMs? The difference will consist only in the fact that in the Socionic Compass the “aristocratic” circles must be used. Examples are given in Figures 10 (for LSI as a representative of the right aristocrats) and 11 (for SLI as a representative of the left aristocrats).

Fig. 10. Use of the Socionic Compass as the Model A of LSI as a representative of the right aristocrats. [Clockwise arrows on the aristocratic circles.]

Fig. 11. Use of the Socionic Compass as the Model A of SLI as a representative of the left aristocrats. [Counter-clockwise arrows.]

We have discussed how in the Socionic Compass one can see the Models A of all the types of information metabolism. For this we needed to attend to which poles of the bipolar traits democrats / aristocrats and right / left the concrete TIM belongs. On this depended on which two of the four circles one must concentrate one’s attention, and also in which direction (clockwise or counter-) the numbers of the socionic functions will increase.

Note that in the Socionic Compass one can see not only the 8-component Models A of all TIMs, but also the 16-component Models B [2]. In this case one will not have to exclude from consideration the two circles on which lie the symbols of the TIMs belonging to one of the poles of the trait democrats / aristocrats. Of course, under these symbols one must now understand not TIMs but IAs with account taken of the signs.

The ninth number should be assigned to the cell containing the symbol differing 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 of the numbers from “1” to “8.” This is connected with the fact that 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 the kindred TIMs differ by the trait right / left, which is precisely what determines the direction of increase of the numbers that interests us.

Conclusion

In this work a new element of socionic infographics has been considered, 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, to each pole of the Socionic Compass is put into correspondence a definite four of TIMs. The uniting proceeds by the principle of the commonality of the macroaspect to which the IAs processed by the first functions of the types entering one four belong.

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

The elements of these groups were represented by their formalized designations, in which their group-theoretic (in the mathematical sense of the word) properties found their reflection. The proposed system of designations makes it possible to perform the binary operations in the indicated groups guided by simple formalized rules, without resorting every time to cumbersome tables, schemes and matrices. In the present work the tables and schemes are used to substantiate these rules, simple in their final form.

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The article was received by the editors on 21.08.2020.