Chapter 3

Chapter 3. A Look at the Bipolar Traits of Types of Information Metabolism from the Standpoint of the Classical Intertype Relations

Generating “New” Groups of Central Sections of the Socion on the Basis of “Old” Bipolar Traits of TIMs

At the time when only one 16-element group of central sections of the socion had been built explicitly (the AR-trait group), the authors of the article [11] specially drew socionists’ attention to the possibility of constructing, by formal-mathematical methods, a simply gigantic number of groups isomorphic to the group already known. But the posing of the task of investigating these groups was judged premature by the article’s authors, since it would have led to the dispersal of the already scanty forces directed at the study of the one known group, which had long been under development.

By what else is the AR-trait group singled out, apart from its long history of development? The analysis of the intertype relations from the position of the AR-trait group led G.R. Reinin to a model of the IR that differs from the classical one (proposed by Aušra Augustinavichiute [1]). It turned out that only to the operators of the subgroup {1, -1, c, -c, m, -m, cm, -cm} ({identity, superego, extinguishment, duality, mirror, conflict, quasi-identity, activation}) can one put into one-to-one correspondence sets of ARTs whose poles coincide for the types united into a pair by the specific operator of that subgroup. In the remaining cases there occurred, in G.R. Reinin’s terminology, a “splitting” of the IR [12, p. 166]. Therefore in his group of operators, in place of the missing 8 operators of the classical IR, new ones appeared, “tied” to “spectra of coinciding traits” (the ARTs being meant).

Later, when the JMT group had been presented explicitly, it was shown that, by analogy with the model of the IR that G.R. Reinin had, one could build yet another “trait-based” model, “tied” to this other group of bipolar traits. In that case one obtained the operators of the subgroup {1, -1, c, -c, 1*, -1*, c*, -c*} ({identity, superego, extinguishment, duality, kindred, business, mirage, semi-duality}) of the group of operators of the classical IR, plus 8 more new operators. As for the “splitting” problem, the asymmetric relations split not into irrational and rational, as with G.R. Reinin, but into democratic and aristocratic. Among the symmetric relations, it was not {1*, -1*, c*, -c*} that underwent “splitting” but {m, -m, cm, -cm} [3]. Recall that the two quadruples of operators just written out in the formalized notations, as well as the quadruple {m*, -m*, cm*, -cm*} of operators of asymmetric IR, are cosets with respect to the centre {1, -1, c, -c} of the group of the classical IR.

It was clear that one could go on breeding “trait-based” models of the IR in this way, “tying” them to the next 16-element group of bipolar traits of TIMs. And in none of them can one obtain operators of asymmetric IR (not in name, but in mathematical essence). For in comparing the poles of bipolar traits in a pair of TIMs, the pair figures as unordered: it does not matter which TIM of the pair is first and which second (which is the initial one for the operator and which the resulting one). Names like rational request could arise only in comparison with the previously proposed classical model of the IR, which led to operators of asymmetric IR automatically. This prompted the thought of looking at the central sections of the socion and their possible groups from the standpoint of the classical model of the IR, which is unique of its kind and is built without being “tied” to any particular group of traits [4].

Let us recall once more that, unlike the “trait-based” models of the IR, the classical model leads to a non-commutative group of operators. There are only 4 operators of the classical IR that commute with all the elements of the group. They constitute a special subgroup, which in mathematics is called the centre of the group. By means of this subgroup the socion is divided into quartets of TIMs to which corresponds the “formula” {-1, c, -c}:

1. {Ne⁺, Se⁺, Ni⁺, Si⁺},
2. {Ne⁻, Se⁻, Ni⁻, Si⁻},
3. {Te⁺, Fe⁺, Ti⁺, Fi⁺},
4. {Te⁻, Fe⁻, Ti⁻, Fi⁻}.

Each of these quartets can be represented as two pairs in which the TIMs are united either by the superego operator (-1), or by the extinguishment operator (c), or by the duality operator (-c). Exactly the same tetrachotomy of the socion results if one uses any pair from the triple of mutually dependent bipolar traits: I1 (irrationals / rationals), I2 (democrats / aristocrats), I3 (right / left).

Supplementing this triple with the trait of existence I0, to whose first pole the whole socion belongs, we obtain the group I = {I0, I1, I2, I3}, by which both the JMT group and the AR-trait group can be decomposed into cosets. The corresponding factor groups we write thus: M = {I, A, B, AB} and R = {I, A, C, AC}.

In Tables 3.1 and 3.2, for the convenience of readers not yet accustomed to the formalized notations of the bipolar traits of the JMT and ART groups, the words (or their abbreviations) fixing the corresponding sections of the socion are given [7].

Table 3.1. Formalization of the notations for the JMTs.

	I	A	B	AB
0	S/∅	stat/dynam	abstr/invol-1	α/γ-1
1	irrat/rat	extr/intr	intern/extern-1	δ/β-1
2	democr/arist	quest/declat	abstr/invol-2	α/γ-2
3	right/left	posit/negat	extern/intern-2	β/δ-2

Table 3.2. Formalization of the notations for the ARTs.

	I	A	C	AC
0	S/∅	stat/dynam	judic/decis	intuit/sens
1	irrat/rat	extr/intr	caref/farsight	tact/strat
2	democr/arist	quest/declat	merry/serious	logicians/ethicists
3	right/left	posit/negat	yield/obstin	constr/emot

The division of the JMT and ART groups into cosets with respect to the subgroup “I” brought out a remarkable peculiarity of that subgroup concerning the intertype relations. The dichotomies of the socion by the traits entering this subgroup can be represented not only as halvings of the 16-element set of TIMs, but also as dichotomies of the 8-element sets of pairs of TIMs of three varieties — the set of superego pairs (-1), the set of extinguishment pairs (c) and the set of dual pairs (-c). As for the remaining cosets with respect to this subgroup, the traits entering them corresponded to sections of the socion representable as dichotomies of only one of the three sets of pairs of TIMs listed.

The set composed of all the elements of the ART and JMT groups is a 24-element one (counting the trait of existence I0). But under the given binary operation this set does not form a group. The group minimal in number of elements that includes both of the order-16 groups under consideration is itself a 32-element one [6]. It is not hard to understand that in decomposing such a group into cosets with respect to the subgroup “I”, besides the cosets already considered, two more will necessarily appear, which it would be entirely natural to denote “BC” and “ABC” (they figure neither in Table 3.1 nor in Table 3.2).

However, as it turns out, into the first of these cosets fall traits of central sections of the socion (8:8), while into the second fall those separating one of the homogeneous quartets of TIMs with “formula” {-1, c, -c} from the rest of the socion — i.e. a non-central section 4:12. So that the class consisting of the non-central traits should stand out by its “three-letter” notation, it was decided to assign, in the ART group, the notation “C” to the class whose traits preserve the dual pairs but split the extinguishment pairs into separate TIMs. In other words, the single letter “C” in the ART group denotes the coset whose traits are orthogonal to the traits of the coset “B” belonging to the JMT group. This is expressed in the fact that the product of any two traits taken one from each of the classes “B” and “C” is itself a central trait.

The central sections of the socion belonging to the class “BC” are not hard to build once one has at one’s disposal the sections from “B” and “C” (or from “AB” and “AC”). One can then obtain two further 16-element groups of central sections, consisting of the quadruples of cosets {I, B, C, BC} and {I, AB, AC, BC}. These groups are isomorphic to the ART and JMT groups. They lack the coset “A”, whose elements can be written out explicitly using the symbols of the superego pairs (-1). By means of these same pairs the sections of the socion entering the class “BC” can be written out:

BC0 = {Ne⁺,Se⁺}, {Ne⁻,Se⁻}, {Te⁺,Fe⁺}, {Ti⁻,Fi⁻} / {Ni⁺,Si⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻}, {Ti⁺,Fi⁺}; BC1 = {Ne⁺,Se⁺}, {Ne⁻,Se⁻}, {Ti⁺,Fi⁺}, {Te⁻,Fe⁻} / {Ni⁺,Si⁺}, {Ni⁻,Si⁻}, {Te⁺,Fe⁺}, {Ti⁻,Fi⁻}; BC2 = {Ne⁺,Se⁺}, {Ti⁻,Fi⁻}, {Ni⁻,Si⁻}, {Ti⁺,Fi⁺} / {Ni⁺,Si⁺}, {Te⁻,Fe⁻}, {Ne⁻,Se⁻}, {Te⁺,Fe⁺}; BC3 = {Ne⁺,Se⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻} / {Ni⁺,Si⁺}, {Ti⁺,Fi⁺}, {Ne⁻,Se⁻}, {Ti⁻,Fi⁻}.

It immediately strikes the eye (if one attends to the colour of the symbols) that the resulting sections are not orthogonal to the section A1 (the extraverts and introverts are distributed unevenly over the poles of these sections). But this should not surprise: it is precisely visual evidence of the fact that in the groups with cosets {I, B, C, BC} and {I, AB, AC, BC} this bipolar trait is absent — as are the remaining “A”-traits, corresponding to sections that are likewise not orthogonal to the “BC”-sections of the socion. The trait A1 is simply more conspicuous in the symbolic notations of the superego pairs (-1), since membership in its poles found reflection in the colour of the symbols adopted in socionics.

And so, the group of cosets {I, A, B, AB, C, AC, BC, ABC} is the factor group of the 32-element group of sections of the socion by the 4-element subgroup “I”. As a generating system for this group of sections the following quintuple can serve: {I1, I2, A0, B0, C0}. The first three traits are common to the ART and JMT groups; the trait B0 enters the JMT group but not the ART group, and C0 conversely. Recall that I1 ⊗ I2 = I3. Hence, if the sections I1 and I2 are given, we can write out I3 explicitly using the simple multiplication rule: to the first pole of the product belong the elements belonging simultaneously to the first poles of the factors, together with those belonging simultaneously to the second poles. Therefore, if (in superego pairs)

I1 = {Ne⁺,Se⁺}, {Ni⁺,Si⁺}, {Ne⁻,Se⁻}, {Ni⁻,Si⁻} / {Te⁺,Fe⁺}, {Ti⁺,Fi⁺}, {Te⁻,Fe⁻}, {Ti⁻,Fi⁻} and I2 = {Ne⁺,Se⁺}, {Ni⁺,Si⁺}, {Te⁻,Fe⁻}, {Ti⁻,Fi⁻} / {Ne⁻,Se⁻}, {Ni⁻,Si⁻}, {Te⁺,Fe⁺}, {Ti⁺,Fi⁺},

then

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

To obtain all the remaining sections it suffices to know, in addition, that

A0 = {Ne⁺,Se⁺}, {Ti⁺,Fi⁺}, {Ne⁻,Se⁻}, {Ti⁻,Fi⁻} / {Ni⁺,Si⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻}, B0 = {Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Te⁺,Ti⁺}, {Te⁻,Ti⁻} / {Se⁺,Si⁺}, {Se⁻,Si⁻}, {Fe⁺,Fi⁺}, {Fe⁻,Fi⁻}, C0 = {Ne⁺,Si⁺}, {Fe⁻,Ti⁻}, {Te⁺,Fi⁺}, {Ne⁻,Si⁻} / {Fe⁺,Ti⁺}, {Se⁻,Ni⁻}, {Se⁺,Ni⁺}, {Te⁻,Fi⁻},

and also that, when the elements of the order-32 group under discussion are multiplied, the letter parts of their formalized notations multiply independently of the digital parts [10].

The digital indices themselves form the 4-element commutative group {0, 1, 2, 3} with the neutral (identity) element “0”. All the elements of this group are involutions — i.e. multiplied by themselves (squared) they give the neutral element. Of course, for writing out the result of a multiplication by enumerating the TIMs (or their pairs) belonging to the poles of the resulting section of the socion, the symbols of the pairs will in some cases have to be replaced by the symbols of the corresponding two individual TIMs. But the final result can always be recorded using pair symbols (extinguishment (c) for the classes “B” and “AC”, duality (-c) for “AB” and “C”, superego (-1) for “A” and “BC”). For the explicit recording of traits of the subgroup “I” and of the coset “ABC”, which contains the non-central traits, the pair symbols of any of the varieties named may be used.

Let us consider two concrete examples. Suppose we know the explicit representations only of the sections of the “basis” quintuple {I1, I2, A0, B0, C0} (all just given), and we must write out explicitly the already verbalized section C3 (yielding / obstinate) and the “nameless” section BC2. Begin with the first: C3 = I3 ⊗ C0 = (I1 ⊗ I2) ⊗ C0. The operation in brackets we have already performed. But the section I3 we found is recorded using the symbols of the superego pairs, while the “basis” section C0 uses the symbols of the dual pairs. For the resulting section C3 a record in dual pairs is expected. Let us rewrite I3 too in dual pairs (for the “I”-sections this is possible!):

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

It remains to multiply by C0 = {Ne⁺,Si⁺}, {Fe⁻,Ti⁻}, {Te⁺,Fi⁺}, {Ne⁻,Si⁻} / {Fe⁺,Ti⁺}, {Se⁻,Ni⁻}, {Se⁺,Ni⁺}, {Te⁻,Fi⁻}. To the first poles of the factors simultaneously belong the dual pairs {Ne⁺,Si⁺} and {Te⁺,Fi⁺}, and to the second poles {Se⁻,Ni⁻} and {Te⁻,Fi⁻}. Consequently, to the first pole of the product belong the pairs {Ne⁺,Si⁺}, {Te⁺,Fi⁺}, {Se⁻,Ni⁻}, {Te⁻,Fi⁻}. To the second pole belong, naturally, the remaining dual pairs. Finally we have:

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

Let us pass to the second example. One could, of course, take the path that suggests itself, using the obvious expression BC2 = I2 ⊗ (B0 ⊗ C0). Let us choose a more exotic path: BC2 = I1 ⊗ (B0 ⊗ C3), since the explicit representation of the trait C3 is already in our possession. We face the multiplication of a “B”-trait, in whose representation extinguishment pairs participate, by a “C”-trait expressed by means of dual pairs. This is an example of precisely that case where the pair symbols will have to be “unfolded,” replaced by the symbols of the two corresponding TIMs. Only then can the simple multiplication rule already known be applied. The result we shall present in “folded” form, using the symbols of the superego pairs. Finally, the multiplication I1 ⊗ BC3 will have to be performed to obtain the explicit expression for BC2 in superego pairs:

B0 = {Ne⁺, Ni⁺, Ne⁻, Ni⁻, Te⁺, Ti⁺, Te⁻, Ti⁻} / {Se⁺, Si⁺, Se⁻, Si⁻, Fe⁺, Fi⁺, Fe⁻, Fi⁻}, C3 = {Ne⁺, Si⁺, Te⁺, Fi⁺, Se⁻, Ni⁻, Te⁻, Fi⁻} / {Fe⁻, Ti⁻, Ne⁻, Si⁻, Fe⁺, Ti⁺, Se⁺, Ni⁺}, BC3 = {Ne⁺, Te⁺, Ni⁻, Te⁻, Fe⁻, Si⁻, Fe⁺, Se⁺} / {Ni⁺, Si⁺, Ti⁺, Fi⁺, Ne⁻, Se⁻, Ti⁻, Fi⁻} = {Ne⁺,Se⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻} / {Ni⁺,Si⁺}, {Ti⁺,Fi⁺}, {Ne⁻,Se⁻}, {Ti⁻,Fi⁻}.

I1 ⊗ BC3 = ({Ne⁺,Se⁺}, {Ni⁺,Si⁺}, {Ne⁻,Se⁻}, {Ni⁻,Si⁻} / {Te⁺,Fe⁺}, {Ti⁺,Fi⁺}, {Te⁻,Fe⁻}, {Ti⁻,Fi⁻}) ⊗ ({Ne⁺,Se⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻} / {Ni⁺,Si⁺}, {Ti⁺,Fi⁺}, {Ne⁻,Se⁻}, {Ti⁻,Fi⁻}) = {Ne⁺,Se⁺}, {Ni⁻,Si⁻}, {Ti⁺,Fi⁺}, {Ti⁻,Fi⁻} / {Ni⁺,Si⁺}, {Ne⁻,Se⁻}, {Te⁺,Fe⁺}, {Te⁻,Fe⁻} = BC2.

As we see, by means of the binary operation introduced by G.R. Reinin [12, p. 147] on the set of bipolar traits, and of the known elements of the ART group and the JMT group, two more groups of the same order can be obtained. All four groups are subgroups of the 32-element group, which can be decomposed with respect to the 4-element subgroup “I” into 8 cosets. One of these classes (“ABC”) consists of non-central sections. One more class (“BC”) consists of central sections that are verbalized neither in the ART group nor in the JMT group.

It might seem that the two “new” 16-element groups, built on the basis of already verbalized sections by means of the known binary operation, have — even from the standpoint of the classical IR — the same structure as the previously built ART and JMT groups. Indeed, each of the four groups includes the subgroup “I”. The class “ABC” enters none of them. Each of the remaining 6 cosets of the 32-element group with respect to the subgroup “I” enters two of the four 16-element subgroups; and the sections of the socion entering each of these 6 classes are characterized by the fact that they do not split the TIMs united in a pair by one of three relations: extinguishment (“B” and “AC”), duality (“AB” and “C”), superego (“A” and “BC”). These three varieties of class are evenly distributed over all four 16-element subgroups. For visual clarity, the 6 cosets in question are presented in Fig. 3.1 as the edges of a tetrahedron. The three edges bounding a particular face of the tetrahedron are the three classes that must be supplemented by the subgroup “I” to yield one of the four 16-element groups of central sections of the socion. Opposite edges of the tetrahedron correspond to those quadruples of central sections of the socion whose explicit representations can be recorded by enumerating the symbols of pairs of one kind (superego, extinguishment or dual).

[Fig. 3.1. A visual representation of the composition of the four existing 16-element subgroups (without the common 4-element subgroup “I”) of the 32-element group of sections of the socion with generating system {I1, I2, A0, B0, C0}.]

So far no essential differences among the 16-element groups of sections of the socion under discussion are visible. They all seem alike, even taking the view from the position of the classical IR into account — though the individual traits entering one group no longer look as “co-ordinate in rank” as they appeared to G.R. Reinin when he built the AR-trait group [12, p. 152]. Let us note in passing that the total number of 16-element groups of central sections of the socion equals 64,864,800 [5]. Which of these groups of sections of the socion deserve attention from the standpoint of the classical intertype relations?

---

Translator’s note: all pair listings in this section (the four central quartets, the BC0–BC3 sections, the superego-pair forms of I1/I2/I3/A0 and the worked C3 and BC2 computations) are reconstructed from the trait definitions and the independent letter/digit multiplication rule; each intermediate product was verified against both computation paths where the text provides them. The ordering of pairs within a pole, and of the four central quartets, is not recoverable from the text layer and follows the canonical order used in earlier chapters.

Dichotomies of the Socion into Halves Homogeneous from the Standpoint of the Classical IR, and “Single-Formula” Tetrachotomies of the Socion with the Superego Operator in the “Formula”

Considering the tetrachotomies of the socion organized by means of pairs of ARTs, G.R. Reinin noticed that not in all cases did one obtain quartets of types in which “all the elements are in an identical intertype situation” [12, p. 172]. Thus arose the division of quartets of TIMs into homogeneous and non-homogeneous — it being understood that the classical, and not the “trait-based,” IR were meant. Clearly one can speak not only of homogeneous quartets of TIMs, but of octets and of pairs as well.

It seems natural, for dividing the socion into homogeneous parts, to bring in the proper subgroups of the group of operators of the classical IR. For now let us consider the “single-formula” divisions. Recall that by a “formula” is meant the set of operators of the classical IR obtained from a subgroup by removing the identity element (the identity operator). Dichotomies of the socion into homogeneous halves are always “single-formula.” And the group of operators of the classical IR has exactly 7 subgroups of order 8. The corresponding dichotomies are already verbalized in the common 8-element subgroup of the ART and JMT groups (the classes “I” and “A”):

1. {1, -1, c, -c, 1*, -1*, c*, -c*} ↔ I1 (irrat / rational),
2. {1, -1, c, -c, m, -m, cm, -cm} ↔ I2 (democr / aristocr),
3. {1, -1, c, -c, m*, -m*, cm*, -cm*} ↔ I3 (right / left),
4. {1, -1, 1*, -1*, m, -m, m*, -m*} ↔ A0 (static / dynam),
5. {1, -1, 1*, -1*, cm, -cm, cm*, -cm*} ↔ A1 (extr / intr),
6. {1, -1, c*, -c*, m, -m, cm*, -cm*} ↔ A2 (quest / declat),
7. {1, -1, c*, -c*, cm, -cm, m*, -m*} ↔ A3 (positiv / negat).

The eighth element in this subgroup is I0 (the trait of existence), to whose first pole the whole socion belongs. Recall that in fixing the order of the poles for all the bipolar traits of TIMs, the count is taken from the ILE ((Ne, Ti)). Note that already at this stage the 16-element groups containing the classes “I” and “A” (i.e. the ART group and the JMT group) have turned out to be singled out against the background of the other groups of the same order entering the 32-element group under consideration.

There exist 15 “single-formula” tetrachotomies of the socion into homogeneous quartets of TIMs (by the number of subgroups of order four), and in 7 cases the superego operator (-1) enters the “formula.” It is precisely in these cases that, should one wish to organize the corresponding divisions of the socion by means of pairs of bipolar traits rather than “formulas,” the traits common to the ART and JMT groups fully suffice. One must bear in mind that one and the same tetrachotomy results from taking any pair of a triple of mutually dependent traits. Therefore to the 3-element “formula” that “works” in each quartet of TIMs under a given tetrachotomy one can put into correspondence a triple of mutually dependent traits. If the “formula” is supplemented by the identity operator (1), and the triple of mutually dependent traits by the trait of existence (I0), the following one-to-one correspondences between subgroups of the respective groups result:

1. {1, -1, c, -c} ↔ {I0, I1, I2, I3},
2. {1, -1, 1*, -1*} ↔ {I0, I1, A0, A1},
3. {1, -1, c*, -c*} ↔ {I0, I1, A2, A3},
4. {1, -1, m, -m} ↔ {I0, I2, A0, A2},
5. {1, -1, cm, -cm} ↔ {I0, I2, A1, A3},
6. {1, -1, m*, -m*} ↔ {I0, I3, A0, A3},
7. {1, -1, cm*, -cm*} ↔ {I0, I3, A1, A2}.

In all the tetrachotomies of the socion listed, the superego pairs (-1) are not split into individual socionic types. It is therefore convenient in this case to regard the socion as an 8-element set of such pairs. And for an 8-element set, any group of central sections of that set maximal in the number of elements likewise contains 8 elements (counting the identity section as the identity element of the group). For the “basis” of an 8-element typology just three bipolar traits suffice; in the present case the triple of traits {I1, I2, A0} is entirely suitable.

For the superego pairs one can introduce operators of inter-pair relations with quite natural formalized notations: {±1, ±c, ±1*, ±c*, ±m, ±cm, ±m*, ±cm*} [9]. The operator ±1 here plays the role of the identity element of the group; it expresses the relations between two identical superego pairs. From the mathematical standpoint this group is the factor group of the group of operators of the classical IR by the subgroup {1, -1}. Note that the group of inter-pair relations under discussion is the only commutative 8-element factor group of the 16-element non-commutative group of operators of the classical IR. The factor groups obtained using the subgroups {1, c} and {1, -c} are non-commutative — and other two-element normal subgroups the group of the classical IR simply does not have.

Replacing the 16-element set of TIMs by the 8-element set of their pairs proves very convenient, since it allows fairly simple infographics to be brought in for the explanations. Indeed, a 3-element basis of an 8-element typology can be represented in a diagram by two straight lines intersecting at a right angle and a circle whose centre coincides with their point of intersection. In Fig. 3.2 (the left diagram) the socion is represented as the 8-element set of superego pairs (-1). There the sections of the socion {I1, I2, A0}, which formed the generating system of the common 8-element subgroup of the ART and JMT groups, are visually presented.

The right diagram is the Cayley graph of the corresponding group of inter-pair relations. As the generating system for this group the triple {±c, ±1*, ±m} has been chosen. The diagrams in the figure are coordinated so that the vertices of the graph are put into correspondence not only with the operators but with the pairs of socionic types. As the “reference point” the pair {Ne⁺, Se⁺} (ILE + SEE) is adopted. With such a coordination of the diagrams and choice of “reference point” it is easy to trace which operators’ action leads to a change of the poles of the traits of the “basis” triple {I1, I2, A0}.

[Fig. 3.2. The generating system of the common order-8 subgroup of the ART and JMT groups as a basis of the typology of superego pairs (-1), and the Cayley graph of the group of inter-pair relations for such pairs.]

All the operators containing the letter “c” in their formalized notation lead to a change of pole of the trait A0 (statics / dynamics). On the left diagram this is reflected in the symbols of the initial and resulting pairs of TIMs lying on opposite sides of the corresponding dividing line (inside and outside the circle). All the operators with the “asterisk” change the pole of the trait I2 (democrats / aristocrats); in that case the symbols of the initial and resulting pairs lie on opposite sides of the vertical dividing line. Finally, the letter “m” entering an operator’s formalized notation indicates a change of pole of the trait I1 (irrationals / rationals); correspondingly, the symbols of the initial and resulting pairs of TIMs end up on opposite sides of the horizontal dividing line.

Note that on the left diagram, besides the “basis” triple {I1, I2, A0}, two more dichotomies are clearly visible: A1 (extraverts / introverts) and I3 (right / left). In the first case the colour of the symbols helps, in the second the sign used as superscript.

Of the dichotomies verbalized by “A”-traits, it remains to see on the diagram only two: A2 (questims / declatims) and A3 (positivists / negativists). These traits can be expressed as products of the traits well represented in Fig. 3.2:

A2 = I2 ⊗ A0 = I3 ⊗ A1; A3 = I3 ⊗ A0 = I2 ⊗ A1.

Let us use, for example, those products in which the trait I3 figures as a factor. We then obtain that to the first pole of the trait A2 — i.e. to the questims — belong the right extraverts and the left introverts: in other words, the 8 TIMs entering the quartet of superego pairs {{Ne⁺,Se⁺}, {Te⁺,Fe⁺}, {Ni⁻,Si⁻}, {Ti⁻,Fi⁻}}. To the first pole of the trait A3 — i.e. to the positivists — belong the right statics and the left dynamics, i.e. {{Ne⁺,Se⁺}, {Ti⁺,Fi⁺}, {Ni⁻,Si⁻}, {Te⁻,Fe⁻}}.

Comparing the right and left diagrams of Fig. 3.2, one can assign to each superego pair (other than {Ne⁺,Se⁺}) the operator of inter-pair relations linking it with the pair {Ne⁺,Se⁺} into a homogeneous quartet of TIMs. It is not difficult to write down the remaining homogeneous quartets of TIMs with the same “formula” (especially as two superego pairs). Finally we have:

1. ±c ↔ {Ni⁺,Si⁺} ↔ {I1, I2, I3} ↔ {{Ne⁺,Se⁺},{Ni⁺,Si⁺}}, {{Ne⁻,Se⁻},{Ni⁻,Si⁻}}, {{Te⁺,Fe⁺},{Ti⁺,Fi⁺}}, {{Te⁻,Fe⁻},{Ti⁻,Fi⁻}};
2. ±1* ↔ {Ne⁻,Se⁻} ↔ {I1, A0, A1} ↔ {{Ne⁺,Se⁺},{Ne⁻,Se⁻}}, {{Ni⁺,Si⁺},{Ni⁻,Si⁻}}, {{Te⁺,Fe⁺},{Te⁻,Fe⁻}}, {{Ti⁺,Fi⁺},{Ti⁻,Fi⁻}};
3. ±c* ↔ {Ni⁻,Si⁻} ↔ {I1, A2, A3} ↔ {{Ne⁺,Se⁺},{Ni⁻,Si⁻}}, {{Ni⁺,Si⁺},{Ne⁻,Se⁻}}, {{Te⁺,Fe⁺},{Ti⁻,Fi⁻}}, {{Ti⁺,Fi⁺},{Te⁻,Fe⁻}};
4. ±m ↔ {Ti⁻,Fi⁻} ↔ {I2, A0, A2} ↔ {{Ne⁺,Se⁺},{Ti⁻,Fi⁻}}, {{Ni⁺,Si⁺},{Te⁻,Fe⁻}}, {{Ne⁻,Se⁻},{Ti⁺,Fi⁺}}, {{Ni⁻,Si⁻},{Te⁺,Fe⁺}};
5. ±cm ↔ {Te⁻,Fe⁻} ↔ {I2, A1, A3} ↔ {{Ne⁺,Se⁺},{Te⁻,Fe⁻}}, {{Ni⁺,Si⁺},{Ti⁻,Fi⁻}}, {{Ne⁻,Se⁻},{Te⁺,Fe⁺}}, {{Ni⁻,Si⁻},{Ti⁺,Fi⁺}};
6. ±m* ↔ {Ti⁺,Fi⁺} ↔ {I3, A0, A3} ↔ {{Ne⁺,Se⁺},{Ti⁺,Fi⁺}}, {{Ni⁺,Si⁺},{Te⁺,Fe⁺}}, {{Ne⁻,Se⁻},{Ti⁻,Fi⁻}}, {{Ni⁻,Si⁻},{Te⁻,Fe⁻}};
7. ±cm* ↔ {Te⁺,Fe⁺} ↔ {I3, A1, A2} ↔ {{Ne⁺,Se⁺},{Te⁺,Fe⁺}}, {{Ni⁺,Si⁺},{Ti⁺,Fi⁺}}, {{Ne⁻,Se⁻},{Te⁻,Fe⁻}}, {{Ni⁻,Si⁻},{Ti⁻,Fi⁻}}.

Thus the examination of the “single-formula” tetrachotomies of the socion into homogeneous quartets of TIMs, in the 7 cases where the superego operator enters the “formula,” has not led us to bipolar traits differing from those that divide the socion into homogeneous octets (from the standpoint of the classical IR).

”Single-Formula” Tetrachotomies of the Socion without the Superego Operator in the “Formula”

Let us now consider the 4 cases of “single-formula” tetrachotomies of the socion where the extinguishment operator (c) enters the “formula” but the superego operator (-1) does not. To these tetrachotomies correspond, in one-to-one fashion, the following subgroups of order 4 of the group of operators of the classical IR:

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

In all these cases the extinguishment pairs (c) are preserved. We shall therefore regard the socion as the 8-element set of such pairs. Decomposing the group of operators of the classical IR into cosets with respect to the subgroup {1, c}, we obtain a factor group whose elements can be understood as operators of inter-pair relations:

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

The operator {1, c} here plays the role of the identity element of the group of inter-pair relations for the extinguishment pairs. Let us verify that this 8-element group is non-commutative. Indeed, for example,

{1*, c*} · {m, cm} = {m*, cm*}, but {m, cm} · {1*, c*} = {-m*, -cm*}.

The binary operation on the set of elements of this group of inter-pair relations is introduced on the basis of the binary operation on the set of operators of the intertype relations. The equalities written down rest on the facts that 1* · m = c* · cm = m* and 1* · cm = c* · m = cm*, and likewise that m · 1* = cm · c* = -m* and cm · 1* = m · c* = -cm*.

It should now be perfectly obvious that this group is isomorphic to, say, the group of operators {1, -1, 1*, -1*, m, -m, m*, -m*} — that is, it is the dihedral group of order 8 (D₈). Such a group is not representable as the direct product of its subgroups. On the other hand, its generating system can be reduced to two mutually non-commuting elements. Correspondingly, the Cayley graph of such a group can be presented as an octagon with alternating edges of two kinds. Fig. 3.3 shows all 4 cases of interest to us of “single-formula” tetrachotomies of the socion into homogeneous quartets of TIMs representable as two extinguishment pairs.

[Fig. 3.3. Four tetrachotomies of the socion into homogeneous quartets of TIMs, organized by means of the “formulas”: {c, 1*, c*}, {c, -1*, -c*}, {c, m, cm}, {c, -m, -cm}.]

It is very important to note that we organize the “single-formula” tetrachotomies under discussion by means of “formulas” consisting of three operators of the classical IR, and not by means of pairs of bipolar traits of TIMs. The “formulas” of interest always contain the operator c, while the other two operators form one of the following pairs: {1*, c*}, {-1*, -c*}, {m, cm}, {-m, -cm}. Only afterwards, in each of the four cases considered, do we pose the question of two orthogonal dichotomies by means of which the same division of the socion into 4 quartets of TIMs could be organized.

The Cayley graphs in the form of octagons make it easy to discover the sought pairs of orthogonal central sections of the socion, representing them as a pair of mutually perpendicular straight lines. And after the poles of the suitable sections have been put into one-to-one correspondence with the quartets of extinguishment pairs (octets of TIMs), there arises the task of verbalizing these sections with bipolar traits.

Look at the upper part of Fig. 3.3. There, in the middle diagram, the symbols of the extinguishment pairs are placed at the vertices of the Cayley graph built for the case where the elements {1*, c*} and {m, cm} are chosen as generators. In the left diagram the pair of orthogonal sections {B0, B1} is chosen so that the socion turns out divided into quartets of TIMs in each of which the “formula” {c, 1*, c*} is realized. In the right diagram the pair {AC0, AC2} corresponds to the “formula” {c, m, cm}.

Can a section from the first pair be orthogonal to a section from the second pair? Since the symbols of the extinguishment pairs are placed identically on all three diagrams, it becomes obvious (visible to the eye!) that the answer to this question is negative. The same question could be posed differently: can the product of two central sections taken from different pairs be a central section? And the negative answer to that question testifies unambiguously that these two pairs of sections cannot enter one and the same group of central sections of the socion.

One can also convince oneself by direct multiplication — of the sections B0 and AC0, say — that they are not orthogonal, i.e. that their product is not a central section of the socion. Comparing the left and right diagrams, it is not hard to write out all the elements belonging to the first pole of the sought product (i.e. of the section ABC0 = AC0 ⊗ B0). Their set consists of the elements belonging simultaneously to the first poles of the factors ({Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Te⁻,Ti⁻}) and the elements belonging simultaneously to the second poles ({Se⁺,Si⁺}, {Se⁻,Si⁻}, {Fe⁻,Fi⁻}). Thus,

ABC0 = {Ne⁺,Ni⁺}, {Ne⁻,Ni⁻}, {Te⁻,Ti⁻}, {Se⁺,Si⁺}, {Se⁻,Si⁻}, {Fe⁻,Fi⁻} / {Te⁺,Ti⁺}, {Fe⁺,Fi⁺}.

A detailed analysis of the upper part of Fig. 3.3 gives hope that readers will have no trouble analyzing the lower part of the same figure. The same tetrachotomy as with the “formula” {c, -1*, -c*} can be organized using the pair of mutually orthogonal central sections {B2, B3}, while to the “formula” {c, -m, -cm} the pair {AC1, AC3} will correspond.

Of course, it is better to put into correspondence with a “formula” consisting of three mutually dependent operators not a pair of sections but a triple of mutually dependent ones — the more so as the third section deserves attention. In the case of the pairs of sections verbalized by JMTs, the third is the section I1 (irrationals / rationals): B0 ⊗ B1 = B2 ⊗ B3 = I1; in the case of the ART group it is I2 (democrats / aristocrats): AC0 ⊗ AC2 = AC1 ⊗ AC3 = I2. Recall that any pair of a triple of mutually dependent sections leads to one and the same tetrachotomy of the socion.

Thus the examination of the “single-formula” tetrachotomies of the socion into homogeneous quartets of TIMs, in the 4 cases where the extinguishment operator (c) enters the “formula” but the superego operator (-1) does not, has led us to 8 bipolar traits differing from those that divide the socion into two equal parts homogeneous from the standpoint of the classical IR. Four of these eight traits (the “B”-traits) enter the JMT group, and the other four (the “AC”-traits) the ART group.

Finally, let us consider the 4 cases of “single-formula” tetrachotomies of the socion where the duality operator (-c) enters the “formula” but the superego operator (-1) does not. To these tetrachotomies correspond, in one-to-one fashion, the following subgroups of order 4 of the group of operators of the classical IR:

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

The group of operators of the classical IR can be decomposed into cosets with respect to the normal subgroup {1, -c}, analogously to how we decomposed by the subgroup {1, c}, and these cosets can be regarded as operators of inter-pair relations for the case of the 8-element set of dual pairs. The resulting factor group is isomorphic to the one obtained in the case of the extinguishment pairs — i.e. it is the dihedral group of order 8 (D₈). By analogy with the diagrams presented in Fig. 3.3 it is not hard to build the diagrams presented in Fig. 3.4. In this case we obtain 4 more JMTs (the “AB”-traits) and 4 ARTs (the “C”-traits). Finally we shall have the following 8 one-to-one correspondences between 4-element groups of operators and of bipolar traits, together with the corresponding quartets of TIMs in the form of dual pairs or extinguishment pairs:

1. {1, c, 1*, c*} ↔ {I0, I1, B0, B1} ↔ {{Ne⁺,Ni⁺},{Ne⁻,Ni⁻}}, {{Se⁺,Si⁺},{Se⁻,Si⁻}}, {{Te⁺,Ti⁺},{Te⁻,Ti⁻}}, {{Fe⁺,Fi⁺},{Fe⁻,Fi⁻}};
2. {1, c, -1*, -c*} ↔ {I0, I1, B2, B3} ↔ {{Ne⁺,Ni⁺},{Se⁻,Si⁻}}, {{Ne⁻,Ni⁻},{Se⁺,Si⁺}}, {{Te⁺,Ti⁺},{Fe⁻,Fi⁻}}, {{Te⁻,Ti⁻},{Fe⁺,Fi⁺}};
3. {1, -c, 1*, -c*} ↔ {I0, I1, AB0, AB1} ↔ {{Ne⁺,Si⁺},{Ne⁻,Si⁻}}, {{Se⁺,Ni⁺},{Se⁻,Ni⁻}}, {{Te⁺,Fi⁺},{Te⁻,Fi⁻}}, {{Fe⁺,Ti⁺},{Fe⁻,Ti⁻}};
4. {1, -c, -1*, c*} ↔ {I0, I1, AB2, AB3} ↔ {{Ne⁺,Si⁺},{Se⁻,Ni⁻}}, {{Se⁺,Ni⁺},{Ne⁻,Si⁻}}, {{Te⁺,Fi⁺},{Fe⁻,Ti⁻}}, {{Fe⁺,Ti⁺},{Te⁻,Fi⁻}};
5. {1, c, m, cm} ↔ {I0, I2, AC0, AC2} ↔ {{Ne⁺,Ni⁺},{Te⁻,Ti⁻}}, {{Se⁺,Si⁺},{Fe⁻,Fi⁻}}, {{Ne⁻,Ni⁻},{Fe⁺,Fi⁺}}, {{Se⁻,Si⁻},{Te⁺,Ti⁺}};
6. {1, c, -m, -cm} ↔ {I0, I2, AC1, AC3} ↔ {{Ne⁺,Ni⁺},{Fe⁻,Fi⁻}}, {{Se⁺,Si⁺},{Te⁻,Ti⁻}}, {{Ne⁻,Ni⁻},{Te⁺,Ti⁺}}, {{Se⁻,Si⁻},{Fe⁺,Fi⁺}};
7. {1, -c, m, -cm} ↔ {I0, I2, C0, C2} ↔ {{Ne⁺,Si⁺},{Fe⁻,Ti⁻}}, {{Fe⁺,Ti⁺},{Se⁻,Ni⁻}}, {{Se⁺,Ni⁺},{Te⁻,Fi⁻}}, {{Te⁺,Fi⁺},{Ne⁻,Si⁻}};
8. {1, -c, -m, cm} ↔ {I0, I2, C1, C3} ↔ {{Ne⁺,Si⁺},{Te⁻,Fi⁻}}, {{Se⁺,Ni⁺},{Fe⁻,Ti⁻}}, {{Te⁺,Fi⁺},{Se⁻,Ni⁻}}, {{Fe⁺,Ti⁺},{Ne⁻,Si⁻}}.

[Fig. 3.4. Four tetrachotomies of the socion into homogeneous quartets of TIMs, organized by means of the “formulas”: {-c, 1*, -c*}, {-c, -1*, c*}, {-c, m, -cm}, {-c, -m, cm}.]

The first 4 tetrachotomies of the socion in this list can be organized by means of pairs of JMTs, and the last 4 by means of pairs of ARTs. In the first case the trait I1 (irrat / rat) necessarily enters the 4-element group of bipolar traits, and in the second case I2 (democr / arist). Thus, if one wants to obtain the “single-formula” tetrachotomies of the socion not by means of the 4-element subgroups of the 16-element non-commutative group of operators of the classical IR, but by means of pairs of bipolar traits of TIMs, traits from both groups (the ARTs and the JMTs) will be required.

”Single-Formula” Octochotomies of the Socion into Homogeneous Pairs of TIMs

By Lagrange’s theorem [2, p. 113], the order of a finite group is a multiple of the order of any of its subgroups. It is for this reason that we speak of the subgroups of the 16-element group of operators of the classical IR as subgroups of orders 8, 4 and 2. By means of these subgroups the socion can be divided into 2, 4 and 8 parts, respectively, homogeneous in the sense of the classical IR. The first two cases we have analyzed in detail, asking by means of which bipolar traits (one and two respectively) the same dichotomies and tetrachotomies of the socion can be organized.

It turned out that all 7 existing dichotomies of the socion into homogeneous halves are verbalized by the traits entering the common 8-element subgroup of the 16-element ART and JMT groups. But for organizing the “single-formula” tetrachotomies into homogeneous quartets of TIMs by means of pairs of central traits, at least 16 more traits are needed — and in each of the groups named only half of them can be found, the other half belonging to the second group. A natural question may arise: will one not have to bring in some other central sections of the socion, not verbalized within these two groups of bipolar traits, in order to organize, by means of triples of independent traits, the “single-formula” octochotomies of the socion into homogeneous pairs of TIMs?

We shall show that the answer to this question is negative. Recall that all the operators of asymmetric IR, which are unsuitable for organizing homogeneous pairs of TIMs, ended up in one class of the 16-element non-commutative group of operators of the classical IR under its decomposition into cosets with respect to the centre of the group. Not counting the identity element of the group (the identity operator, 1), the operators of symmetric IR number 11. But they cannot be called co-ordinate in rank.

A special place among the operators of symmetric IR is occupied by the superego operator [9]. Of the octochotomy of the socion into superego pairs we have already spoken. In Fig. 3.2 one of the possible suitable triples of independent traits, {I1, I2, A0}, was presented, which can secure such an octochotomy of the socion. The use of the formalized notations for the traits of the ART and JMT groups allows one, without particular difficulty, to indicate 4 more traits whose poles coincide for the TIMs of a superego pair. Indeed, it is easy in such notations to write down the group of bipolar traits for which the indicated triple is a generating system. Besides this triple and the trait of existence (I0), it will include the traits of the quadruple {I3, A1, A2, A3}. Thus the resulting group is the common 8-element subgroup of the ART group and the JMT group. Any 3-element generating system of this group (of the 28 possible) will do as a triple of traits for organizing the division of the socion into 8 superego pairs.

The next two distinguished operators of the classical IR are the operators of extinguishment (c) and duality (-c). For the division of the socion into extinguishment pairs (c), from the JMT group the triple {I1, I2, B0} is suitable, generating a group that contains in addition the quadruple {I3, B1, B2, B3}, not counting I0. And from the ART group the triple {I1, I2, AC0} is suitable, generating a group that contains in addition the quadruple {I3, AC1, AC2, AC3}. Since any suitable triple of independent traits in this case does not belong wholly to the common subgroup of the ART and JMT groups, but relates either to the one or to the other, the total number of variants turns out to equal 56. The situation with the division of the socion into dual (-c) pairs is entirely analogous. One can name at once one suitable triple from each of the two groups: {I1, I2, AB0} (from the JMT group) and {I1, I2, C0} (from the ART group), which generate in those groups the corresponding 8-element subgroups. Besides I0, in the case of the JMT group such a subgroup will also include the quadruple {I3, AB1, AB2, AB3}, and in the case of the ART group {I3, C1, C2, C3}.

The remaining 8 operators can be divided into two parts, coinciding with the two cosets ({1*, -1*, c*, -c*} and {m, -m, cm, -cm}) under the decomposition of the group of operators of the classical IR by its centre. In the first case the suitable triples of bipolar traits are to be sought in the JMT group, and in the second in the ART group. Let us indicate for each operator one triple (of the 28 possible) generating an 8-element group in which 27 more suitable triples can be found. In the following list, after each suitable triple of independent traits comes the quadruple of traits that enters the 8-element group generated by that triple:

1* → {I1, A0, B0} → {A1, B1, AB0, AB1}, -1* → {I1, A0, B2} → {A1, B3, AB2, AB3}, c* → {I1, A2, B0} → {A3, B1, AB2, AB3}, -c* → {I1, A2, B2} → {A3, B3, AB0, AB1}, m → {I2, A0, AC0} → {A2, AC2, C0, C2}, -m → {I2, A0, AC1} → {A2, AC3, C1, C3}, cm → {I2, A1, AC0} → {A3, AC2, C1, C3}, -cm → {I2, A1, AC1} → {A3, AC3, C0, C2}.

One should note that, having written the lines for 1* and m, it is not hard to obtain the remaining 6 of the 8 given. This is connected with the fact that the operators of the triple {-1*, c*, -c*} can be obtained by multiplying the operators of the triple {-1, c, -c} by the operator 1*, and multiplying the operators of the same triple by the operator m gives the triple {-m, cm, -cm}. It is here assumed, of course, that the analogous lines for the operators of the triple {-1, c, -c} have been obtained by us in advance.

Let us demonstrate on a concrete example how this is done. Suppose we must obtain the 8-element subgroup corresponding to the operator cm, which is representable as the product of the mutually commuting operators m and c. Since to the operator m corresponds a subgroup from the ART group, for the operator c a subgroup from the same group must be taken:

c → {I0, I1, I2, I3, AC0, AC1, AC2, AC3} and m → {I0, I2, A0, A2, AC0, AC2, C0, C2}.

To the operator cm corresponds the 8-element subgroup of the ART group comprising the operators that are simultaneously elements of both subgroups given, together with the operators entering neither of them — that is,

cm → {I0, I2, AC0, AC2, A1, A3, C1, C3}.

Note also that for the duality operator (-c) the corresponding subgroup could have been found by the same algorithm from the known subgroups for the operators of extinguishment (c) and superego (-1), which likewise commute with each other.

Thus, as we have had occasion to convince ourselves, the organization of the “single-formula” octochotomies of the socion into homogeneous pairs of TIMs by means of triples of independent traits did not require central sections that were not verbalized by traits of the ART and JMT groups. Let us repeat once more that the “single-formula” divisions of the socion into parts homogeneous from the standpoint of the classical IR are quite naturally carried out by means of the proper subgroups of the corresponding 16-element non-commutative group of operators (or by means of the “formulas” obtained from them by discarding the identity element). But if one wants to carry out these divisions by means of bipolar traits (one, two or three, for organizing the divisions of the socion into two, four or eight parts respectively), then traits from the ART group and the JMT group will be required.

Conclusion to Chapter 3

The socion, as a 16-element set of TIMs, formally admits more than 30 thousand possible partitions into two parts. The central dichotomies of the socion alone (8 types in each part) number 6,435 [12, p. 234]. Which sections of this set deserve the closer attention of socionists? What is the criterion of choice? How should one select verbal or formalized notations for those sections which for some reason have drawn attention, so that these notations should be convenient at least at the stage of research work and in discussing its current results among the researchers themselves?

The present chapter has considered the approach to seeking answers to these questions that is directly connected with the application in socionics of the mathematical theory of groups. To the classical model of the intertype relations proposed by the founder of socionics, Aušra Augustinavichiute, corresponds a 16-element non-commutative group of operators, using which one can, from the model of one TIM, reconstruct the models of the remaining 15 TIMs. The classical model of the IR is not tied to any particular 16-element commutative group of central sections of the socion. This allows one to look at the simply gigantic set of these groups from a single point of view.

By means of the subgroups of the group of operators of the classical IR one can partition the socion into homogeneous (in G.R. Reinin’s terminology) parts, in which all the TIMs “are in an identical intertype situation” [8]. Interest in such partitions of the socion gives occasion for the verbalization of those sections of the socion by means of which the partition that has interested the researcher can be obtained. On the other hand, the introduction of a binary operation on the set of sections of the socion gives occasion to attend to those sections that result as “products” of sections already verbalized.

Four independent central sections of the socion that are pairwise orthogonal (i.e. whose products are also central sections) partition the socion into individual types. Such a quadruple of central sections is a generating system of a 16-element group of central sections, which is a subgroup of the group of all possible sections of the socion. So far only two such 16-element groups of central sections have been verbalized by “bipolar traits” (the ART group and the JMT group). These two groups have a common 8-element subgroup. For the elements of these two groups convenient formalized notations have been proposed, making it possible to perform the binary operation quickly. Multiplying the elements of these two groups, one can obtain 8 more sections of the socion, of which 4 are central. The formalized notations for these 8 “new” sections were obtained in a natural way from the notations of the ARTs and the JMTs. In the resulting 32-element group there are 4 subgroups of order 16.

In this chapter these four mutually isomorphic 16-element subgroups of the constructed 32-element group of sections of the socion were compared from the standpoint of the classical IR. Along the way it was shown that from this standpoint the ART and JMT groups turn out to be singled out not only in comparison with the two other subgroups of the 32-element group we constructed, but also against the background of the enormous number of subgroups isomorphic to them within the group of all possible sections of the socion, whose mathematical properties G.R. Reinin was the first to study.

The clarification of the connections between the traits characterizing the communicants and the intertype relations between them was long complicated by the absence of suitable notations for the IR operators and the bipolar traits of TIMs that would reflect the mathematical properties of the corresponding groups. That is why the present monograph devotes so much attention to the analysis of concrete examples demonstrating the convenience of the proposed system of notation for the mental operations involved in the study of socionic models.

References for Chapter 3

1. Augustinavichiute A. Sotsionika [Socionics]. — Moscow: Chernaya belka, 2008. — 568 pp.
2. Grossman I., Magnus W. Gruppy i grafy [Groups and Their Graphs]. — Moscow: Mir, 1971. — 246 pp.
3. Minaev Yu.P. The problem of the “splitting” of intertype relations by spectra of coincidences of the poles of bipolar traits // SMiPL. — 2014. — №3. — pp. 30–37.
4. Minaev Yu.P. From intertype relations to two groups of bipolar traits // SMiPL. — 2014. — №6. — pp. 5–17.
5. Minaev Yu.P., Datsenko I.P., Shepel K.S. Groups of central sections of a 2ⁿ-element set and bipolar traits // SMiPL. — 2016. — №3. — pp. 41–51.
6. Minaev Yu.P. From the 16-element group of operators of the classical intertype relations to the 32-element group of traits of the socionic types // SMiPL. — 2016. — №5. — pp. 50–59.
7. Minaev Yu.P., Datsenko I.P., Statiev V.D. A comparative analysis of three models of the intertype relations in the socion // SMiPL. — 2018. — №1. — pp. 39–53.
8. Minaev Yu.P., Reinin G.R. Mathematical modelling of intertype relations // SMiPL. — 2018. — №6. — pp. 48–61.
9. Minaev Yu.P., Datsenko I.P., Safronyuk A.S. Remarkable properties of the superego operator // SMiPL. — 2019. — №2–3. — pp. 33–40.
10. Minaev Yu.P., Datsenko I.P., Pinda M.V. Simple rules for multiplying bipolar traits // PiSMO. — 2017. — №11–12. — pp. 25–35.
11. Osipov A.V., Reinin G.R. The structure of the set of central traits // SMiPL. — 2005. — №5. — pp. 48–54.
12. Reinin G. Tayny tipa. Modeli. Gruppy. Priznaki [Secrets of Type. Models. Groups. Traits]. — Moscow: Chernaya belka, 2010. — 296 pp.

---

Translator’s note: as in §1, all pair listings (the seven superego-pair tetrachotomies, the eight tetrachotomy correspondences, and the ABC0 computation) are reconstructed from the operator orbits and trait definitions; the reference pair {Ne⁺, Se⁺} is fixed by the text (“ILE + SEE”), and each quartet was verified against its “formula.” Pair ordering within listings follows the canonical order of earlier chapters.