Simplifying the Structure of the Intertype Relations Table Using Formalised Notation for Their Operators
Yu. P. Minaev, I. P. Datsenko, E. G. Shevchenko Psychology and Socionics of Interpersonal Relations, No. 1–2, 2016, pp. 50–57 UDC 159.9.075+159.923 (Section: Mathematical Methods in Socionics)
Abstract
A comparative analysis is carried out of three published variants of the intertype relations table in which the authors use differently ordered lists of socionic types. It is shown that even minor changes in the sequence of types can strongly affect how well-structured the table is. The question is considered of how constructing the table on the basis of a normal subgroup of the group of intertype relation operators can reduce the number of distinct operators in a single 4×4 block.
A new variant of the intertype relations table in the socion is presented. The table possesses “block symmetry” with respect to its center. In each 4×4 block only 4 distinct intertype relation operators occur. The variety of blocks is limited to four kinds. The use of formalized notation for the intertype relation operators makes the simplicity of the table’s structure vividly apparent.
Keywords: socionics, intertype relations table, group of intertype relation operators, normal subgroups, commutation relations.
Translator’s legend
The paper uses Russian two-letter abbreviations for the 16 classical intertype relations (IRs). They are kept as in the original, with this key:
| Abbr. | Relation | Operator |
|---|---|---|
| ТО | identity | 1 |
| СЭ | super-ego | –1 |
| РО | kindred | 1* |
| ДЕ | business | –1 |
| ПО | extinguishment (contrary) | c |
| ДУ | duality | –c |
| МИ | mirage | c* |
| ПД | semi-duality | –c* |
| ЗЕ | mirror | m |
| КФ | conflict | –m |
| Р– | supervision (supervisee side) | m* |
| Р+ | supervision (supervisor side) | –m* |
| КВ | quasi-identity | cm |
| АК | activation | –cm |
| З+ | benefit (benefactor side) | cm* |
| З– | benefit (beneficiary side) | –cm* |
TIM acronyms are transliterated in the standard way (ИЛЭ = ILE, СЭИ = SEI, etc.).
Introduction
The formalized notation for intertype relation (IR) operators that we proposed earlier [6] makes it possible to dispense entirely with the famous Augustinavichiute–Liashkiavichius table. However, to become accustomed to using the new IR operator notation for socionic “computations,” practice is required. For such practice it is useful to employ exercises in composing one’s own IR tables, in which the number and sequence of the types of information metabolism (TIMs) are varied.
Starting from short fixed sequences, one can progress to creative exercises in independently ordering the entire set of TIMs so that the table acquires a meaningful structure. Such exercises help not only to acquire the skill of rapidly performing operations with the formalized IR operator notation, but also to become better acquainted with the properties of the noncommutative 16-element group D8×C2, of which the group of classical (per Augustinavichiute) IR operators is a concrete representative. Moreover, by working on composing one’s own tables, one can better understand the construction of tables proposed by other authors, and also learn to find relatively quickly the unfortunate “blunders” that are sometimes encountered there.
New formalized notation for the IR operators
In article [6], formalized notation for the IR operators was proposed. This notation reflects the transformation rules of the abbreviated symbolic designations of TIMs under the action of the corresponding operators. The basic idea was that the whole set of operators should be divided into those under whose action the pole of the bipolar trait irrationality/rationality is preserved, and those under whose action it changes.
In the first case one had to compare the symbol of the “resulting” TIM with the symbol of the “initial” TIM; in the second case — not with the symbol of the “initial” TIM, but with the symbol of its mirror partner. What was tracked was the coincidence/non-coincidence of the color, sign, and shape of the symbols. Only two kinds of shape change were admitted: triangle ↔ circle, square ↔ “little couch.”
The correspondence between the “old” notation (as in book [1]) and the new formalized notation is given in Table 1. The same table records the information on coincidence/non-coincidence of the color, sign, and shape of the abbreviated symbolic designation of the “resulting” TIM when compared with the “initial” TIM (for operators preserving the pole of the irrationality/rationality trait), or with the mirror partner of the “initial” TIM (otherwise).
Table 1. The principle behind the formalized notation for IR operators.
| ТО | СЭ | РО | ДЕ | ПО | ДУ | МИ | ПД | ЗЕ | КФ | Р– | Р+ | КВ | АК | З+ | З– | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| coincidence | (with the symbol of the identical) | (with the symbol of the mirror partner) | ||||||||||||||
| of color | + | + | + | + | – | – | – | – | + | + | + | + | – | – | – | – |
| of sign | + | + | – | – | + | + | – | – | + | + | – | – | + | + | – | – |
| of shape | + | – | + | – | + | – | + | – | + | – | + | – | + | – | + | – |
| 1 | –1 | 1* | –1* | c | –c | c* | –c* | m | –m | m* | –m* | cm | –cm | cm* | –cm* |
As the generating elements of the whole group of classical (per Augustinavichiute) IR operators, three operators of symmetric IRs are taken: 1* (РО), m (ЗЕ) and c (ПО). Recall that the first two of these are the generating elements of a subgroup of order 8, whose operators preserve the pole of the bipolar trait statics/dynamics. Indeed, (1*)² = m² = 1; 1* · m = m*; (m*)² = –1; –1 · m = –m; –1 · m* = –m*; –1 · 1* = –1*. This subgroup is the noncommutative group D8 (the dihedral group of order 8). When the proposed IR operator notation is used, it suffices to remember just one commutation relation:
1 · m = –m · 1.**
The last of the generating elements of the whole 16-element group (the operator c) commutes with all the operators of the group, and its square equals the unit element (the identity operator). Therefore the whole group has the form of the direct product D8×C2. Multiplying in turn the operators preserving the pole of the bipolar trait statics/dynamics by the operator c, we obtain all the remaining 8 elements of the group of classical IR operators.
Note that besides the operator c, three more operators commute not only among themselves but with all the other elements of the group. All four operators possessing this property form the so-called center of the group: {1, –1, c, –c} ({ТО, СЭ, ПО, ДУ}).
Normal subgroups of order 4
The use of the new formalized notation for the IR operators makes it possible to find, without particular difficulty, all 15 subgroups of order 4:
- {1, –1, c, –c}
- {1, –1, 1, –1**}
- {1, –1, c, –c**}
- {1, –1, m, –m}
- {1, –1, m, –m**}
- {1, –1, cm, –cm}
- {1, –1, cm, –cm**}
- {1, c, 1, c**}
- {1, c, –1, –c**}
- {1, c, m, cm}
- {1, c, –m, –cm}
- {1, –c, 1, –c**}
- {1, –c, –1, c**}
- {1, –c, m, –cm}
- {1, –c, –m, cm}
It is easy to see that all possible 4-element subgroups have been listed. But only 7 of the 15 subgroups of order 4 are normal (or invariant, or self-conjugate). If a subgroup is not normal, then there necessarily exists an element of the group, not belonging to that subgroup, such that multiplying the elements of the subgroup by it yields different sets depending on whether the multiplication was performed on the left or on the right. For example, the “Quadra” subgroup {1, –c, m, –cm} ({ТО, ДУ, ЗЕ, АК}) is not normal. Indeed, multiplying by 1* (РО), we obtain either {1, –c, m*, –cm***} ({РО, ПД, Р–, З–}) or {1, –c, –m*, cm***} ({РО, ПД, Р+, З+}). An analogous situation arises when the elements of the “Quadra” subgroup are multiplied by –1, m, –m*, c*, –c*, cm*, –cm***. But when the elements of this same subgroup are multiplied by –1, c, –m, –cm, the answer does not depend on whether we multiply on the right or on the left.
What consequences does the normality or non-normality of a subgroup of order 4 have for the IR table? Suppose the socion is divided into 4 parts, in each of which the corresponding set of three IRs is realized (all elements of the subgroup except the unit). Then in the IR table the TIMs are grouped by the resulting fours, and the “body” of the table, filled with IR operators, is partitioned into 16 blocks 4×4. How many distinct operators can be found in each block? If the partition was carried out by means of a normal subgroup, then in all blocks only 4 distinct operators will occur. And the varieties of blocks by composition (disregarding the order of arrangement of individual elements) will also number four. This is a direct consequence of the fact that a normal subgroup partitions the group unambiguously into cosets, one of which is the subgroup itself, since there is no distinction between left and right cosets.
It is another matter if we have a partition of the socion by means of the subgroups numbered 8–15 in our list. These are not normal. What results from this is well seen in the example of the famous Augustinavichiute–Liashkiavichius table. In half of the blocks there are 4 distinct IR operators each, and in the other half — 8 each. The second variant corresponds to the set of possible relations with representatives of neighboring quadras. The right and left dual pairs of one and the same quadra make their own “contributions” (4 operators each) to the total number of distinct operators found in the block corresponding to IRs with a specific neighboring quadra. But the full set of distinct operators does not depend on whether we speak of IRs with representatives of the preceding quadra or the succeeding one.
Let us dwell on this last assertion. In I. N. Kalinauskas’s book there is a diagram captioned “Structure of interquadral relations” [2], on which the following labels for quadras figure: “my quadra,” “the opposite quadra,” “junior relatives,” “senior relatives.” One might think that the last two labels refer to the two quadras neighboring “mine,” with the “junior relatives” belonging to the succeeding quadra and the “senior” ones to the preceding. Note that according to the diagram given in the book, the “junior relatives” are obtained from “my” TIM by the action of the operators m* (Р–), –cm* (З–), –1* (ДЕ), c* (МИ), and the “senior” ones by means of the operators –m* (Р+), cm* (З+), 1* (РО), –c* (ПД). Is this supposition about the link between the words “junior” and “succeeding,” and likewise “senior” and “preceding,” justified? It is justified only for the right TIMs; for the left ones everything is exactly the other way around.
If one compares, by their set of IR operators, the succeeding and the preceding quadra in the evolutionary order (…→α→β→γ→δ→α→…) relative to a given one, they cannot be distinguished! In both cases it will be the same set of 8 elements. Therefore all IR tables based on the quadra-wise division of the socion will contain 4 blocks in which only the operators 1 (ТО), –c (ДУ), m (ЗЕ) and –cm (АК) can be found; 4 blocks with the operators –1 (СЭ), c (ПО), –m (КФ) and cm (КВ); and each of the remaining 8 blocks will contain all 8 remaining operators. The order of arrangement of these operators within the blocks, however, may differ.
Dependence of the structure of the “quadra-wise” IR table on the order of TIMs within the quadras
Consider, by way of example, the tables given in the three books [1; 3; 7]. In all three cases the order of the quadras is the same, the dual pairs are not split apart, and an extravert comes first in each. Moreover, in the first two quadras the order of TIMs coincides completely across all three sources, and in these quadras the right diad comes first, followed by the left.
Book [1] gives the famous Augustinavichiute–Liashkiavichius table, where the third and fourth quadras also begin with right diads. Book [7] gives the Liashkiavichius–Churyumov table, where the last two quadras begin with left diads, while in book [3] only in the third quadra is the left diad moved to first place; in the fourth quadra the order of diads is the same as in the first two quadras. How did such differences in the arrangement of diads in the last two quadras affect the IR tables?
The composition of the blocks is of course the same in all three cases, since quadra-wise division of the socion is used everywhere. However, the order of arrangement of the operators within the blocks differs substantially. This does not apply to the blocks located on the main diagonal of the table, which collect the operators corresponding to intraquadral relations. A permutation of the right and left diads cannot change the order of operators in such blocks. Also identical in all three compared cases are the blocks corresponding to the “interactions” of the first two quadras, since for these quadras the order of TIMs was everywhere the same.
Should any reader of this article wish to verify directly the coincidence of the indicated blocks across all three sources, they should be prepared for certain difficulties. First, the authors of these sources use different notations for the IR operators. Second, the Liashkiavichius–Churyumov table published in [7] contains obvious “blunders,” which were pointed out to Semyon Ivanovich, and a corrected variant of this table can now be found on his personal website.
It should be said that the Liashkiavichius–Churyumov table posted on the site is, moreover, in color. In it, identical blocks are highlighted in one color. Therefore the “block symmetry” of this table relative to its center immediately catches the eye. Coloring the table required six colors (or shades of one color). Note that some of the differing blocks in this table can easily be obtained from one another. In one case the transition operator is the super-ego operator (–1, СЭ), in the other — quasi-identity (cm, КВ). In the latter case the corresponding blocks contain only operators that commute with the quasi-identity operator, so no problems arise with a possible difference between the results of multiplication on the left and on the right. The super-ego operator is “unproblematic” in this sense: it commutes with all the IR operators.
The most “poorly surveyable” of the compared tables is the one from book [3, p. 104]. In it one can count seven distinct 4×4 blocks. We were unable to discover any symmetry or orderliness in the arrangement of the blocks. Perhaps this was one of the first IR tables, produced when one wanted to obtain at least some reference table recording the result that followed from Aushra Augustinavichiute’s idea of classifying IRs by the connections between the socionic functions processing the same informational aspect (IA) in two communicants.
Compared with the table from book [3], the Liashkiavichius–Churyumov table has, of course, a clear advantage in terms of structuredness. And one gets the impression that Semyon Ivanovich, in speaking of his “Theorem on the strict ordering of the set of types in the socion,” is comparing his own table precisely with this one. After all, he indicates that the only thing distinguishing his “mathematically proven” order of TIMs from the order fixed in V. Liashkiavichius’s table is the interchange of the right and left dual pairs in the fourth quadra [8, p. 387]. But by this very feature his order differs from the order in the table from N. N. Medvedev’s book [3].
In comparison with the famous table which can now be found on the last page of every issue of the socionics journals published by the International Institute of Socionics, Semyon Ivanovich has the dual diads interchanged not only in the fourth but also in the third quadra. That in V. Liashkiavichius’s table all quadras begin with right diads, anyone can verify by opening p. 226 of G. A. Shulman’s book [9], where he reproduces a copy of the handwritten variant of the table which Aushra Augustinavichiute’s “Kaunas pupils” presented to her, and on which she left her autograph.
V. V. Gulenko’s book [1] gives this very same famous Augustinavichiute–Liashkiavichius IR table, but with altered notation for the IR operators, with different words designating the TIMs, and with the “body” of the table reflected about the main diagonal. The latter is connected with changing the question from “Who is he to me?” to “Who am I to him?“. In the notation used by V. V. Gulenko for the IR operators, the operators of asymmetric IRs are more clearly visible against the background of the symmetric ones. Moreover, in his variant of the Augustinavichiute–Liashkiavichius table, the blocks in which 8 distinct operators can be found are highlighted by shading. It is immediately seen that there are only two kinds of such blocks. The unshaded blocks are those containing 4 distinct operators each — and of these there are likewise two kinds. Thus the total number of block varieties in the table turns out to be four.
If one analyzes how the various unshaded blocks are related to one another, one notices that they pass into one another by means of the super-ego operator (–1, СЭ). The various shaded blocks are related by the same operator! The analyzed table has no “block symmetry” about the center such as the Liashkiavichius–Churyumov table possesses, but it has another interesting feature.
If the “body” of this table is mentally cut into horizontal strips 4×16, and the short sides of each strip are joined so that the “text” ends up on the outer side of the resulting cylindrical surface, then the four cylinders cannot be distinguished from one another by relying only on the “text” written on their surfaces, ignoring the places where they are “glued.” The indicated features of the famous Augustinavichiute–Liashkiavichius IR table become still more noticeable if we use the formalized notation for the IR operators proposed by us (Table 2).
Table 2. The Augustinavichiute–Liashkiavichius IR table composed using the formalized operator notation.
Read as the equation y = Ax: the “initial” TIM (x) is in the header row, the “resulting” TIM (y) in the left column; the cell gives the operator A.
| y=Ax → | ILE | SEI | ESE | LII | EIE | LSI | SLE | IEI | SEE | ILI | LIE | ESI | LSE | EII | IEE | SLI |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ILE | 1 | -c | -cm | m | cm* | -m* | -1* | c* | -1 | c | cm | -m | -cm* | m* | 1* | -c* |
| SEI | -c | 1 | m | -cm | -m* | cm* | c* | -1* | c | -1 | -m | cm | m* | -cm* | -c* | 1* |
| ESE | -cm | m | 1 | -c | 1* | -c* | -cm* | m* | cm | -m | -1 | c | -1* | c* | cm* | -m* |
| LII | m | -cm | -c | 1 | -c* | 1* | m* | -cm* | -m | cm | c | -1 | c* | -1* | -m* | cm* |
| EIE | -cm* | m* | 1* | -c* | 1 | -c | -cm | m | cm* | -m* | -1* | c* | -1 | c | cm | -m |
| LSI | m* | -cm* | -c* | 1* | -c | 1 | m | -cm | -m* | cm* | c* | -1* | c | -1 | -m | cm |
| SLE | -1* | c* | cm* | -m* | -cm | m | 1 | -c | 1* | -c* | -cm* | m* | cm | -m | -1 | c |
| IEI | c* | -1* | -m* | cm* | m | -cm | -c | 1 | -c* | 1* | m* | -cm* | -m | cm | c | -1 |
| SEE | -1 | c | cm | -m | -cm* | m* | 1* | -c* | 1 | -c | -cm | m | cm* | -m* | -1* | c* |
| ILI | c | -1 | -m | cm | m* | -cm* | -c* | 1* | -c | 1 | m | -cm | -m* | cm* | c* | -1* |
| LIE | cm | -m | -1 | c | -1* | c* | cm* | -m* | -cm | m | 1 | -c | 1* | -c* | -cm* | m* |
| ESI | -m | cm | c | -1 | c* | -1* | -m* | cm* | m | -cm | -c | 1 | -c* | 1* | m* | -cm* |
| LSE | cm* | -m* | -1* | c* | -1 | c | cm | -m | -cm* | m* | 1* | -c* | 1 | -c | -cm | m |
| EII | -m* | cm* | c* | -1* | c | -1 | -m | cm | m* | -cm* | -c* | 1* | -c | 1 | m | -cm |
| IEE | 1* | -c* | -cm* | m* | cm | -m | -1 | c | -1* | c* | cm* | -m* | -cm | m | 1 | -c |
| SLI | -c* | 1* | m* | -cm* | -m | cm | c | -1 | c* | -1* | -m* | cm* | m | -cm | -c | 1 |
| ПД | РО | Р– | З– | КФ | КВ | ПО | СЭ | МИ | ДЕ | Р+ | З+ | ЗЕ | АК | ДУ | ТО |
We did not use the labels customary for this table, such as “I,” “he,” “he — to me.” Instead, the equation y = Ax is written, with arrows indicating that the variants of the “initial” TIM (x) should be sought in the row above the “body” of the table, and the variants of the “resulting” TIM (y) in the column to the left of the “body.” The TIMs are designated by the abbreviated symbolic notation coinciding with the symbols of the IAs “processed” by the first functions, with an indication of the sign by which the IA of the second function can be recognized. Note that from the combination of the first function’s IA symbol and the sign, the TIM’s membership in a particular quadra (and also in a particular “Club”) is easily determined.
As was shown at the beginning of the article, the symbolic designation of the operator A is determined without particular difficulty from the abbreviated symbolic designations of the “initial” (x) and “resulting” (y) TIMs. So anyone who wishes may independently check whether an accidental error has crept into the table. Without the formalized notation this is far harder to do. And the real example we gave with the Liashkiavichius–Churyumov table published in [7] is concrete confirmation of this.
Note the row beneath the “body” of Table 2. It duplicates the bottom row of the table’s “body,” “translated” into the notation adopted in [1]. The operators written in this row essentially answer the question: “What is СЛИ (SLI) to the TIMs indicated in the row located above the ‘body’ of the table?” Let us see what this TIM is to the representatives of the first quadra. As we see, for the right dual pair it is, in I. N. Kalinauskas’s terminology, a “senior relative” (semi-dual and kindred respectively), and for the left pair a “junior relative” (supervisee-adjacent and beneficiary-adjacent respectively). Yet for SLI all four TIMs of the first quadra are supposed to count as “senior” relatives (ПД, РО, Р+, З+). Thus a logical contradiction results! And this is precisely connected with the fact that the “Quadra” subgroup of IR operators is not normal. Left and right TIMs play fundamentally different roles in the relay of quadras. The right ones receive the baton from the preceding quadra, while the left ones pass it to the succeeding one. Left and right TIMs of the same quadra have different sets of IRs with the TIMs of a specific neighboring quadra.
This ring structure of the socion does find some reflection in the TIM sequence used in the Augustinavichiute–Liashkiavichius IR table. When the abbreviated symbolic TIM designations are used, it is easier to notice that on either side of the boundary between neighboring quadras stand “related” dual diads. Together they form a four corresponding to the subgroup of IR operators that has received the name “Square”: {1, 1, –c, –c**} ({ТО, РО, ДУ, ПД}). Thus every dual pair belongs both to its quadra and to its square, which joins two neighboring quadras.
Formally, of course, one could say that in the ring of the socion the quadras join the squares. This formal parity between quadras and squares is especially well seen in the so-called “Structural formula of the socion” [4; 5]. We shall not go deeper into this topic now. Let us note only that if we “ringed” the TIM sequence adopted in the Augustinavichiute–Liashkiavichius IR table and then broke the ring again, starting the count not from ILE but from IEE, and “went” in the same direction, we would obtain a square-wise division instead of a quadra-wise one. And an IR table with a similar structure would result, since “Square” as a subgroup of the IR operator group (and not as a four of TIMs), exactly like the “Quadra” subgroup, is not normal (or invariant, or self-conjugate).
As we have already seen, the ring structure of the socion is not very well visible in the IR table even with a TIM order favorable for it (as in the Augustinavichiute–Liashkiavichius table). For demonstrating the ring structure, the planar undirected Cayley graph of the noncommutative group of classical IR operators, which is isomorphic to the abstract group D8×C2 [5], is better suited. In this connection the question arises: does it make sense, when composing an IR table — if one really wants to have one — to insist on tying the TIM sequence to the quadras? Perhaps, if a normal subgroup is taken as the basis, a more structured table can be built, since in all 4×4 blocks, as we have already noted, there will be only 4 distinct operators? For our IR operator notation, the TIM sequence used in Table 3 proved felicitous. As the normal subgroup by means of which the socion is divided into four fours, the “Socionic temperament” subgroup was taken: {1, –1, 1, –1**} ({ТО, СЭ, РО, ДЕ}). Its cosets are: {c, –c, c, –c**} ({ПО, ДУ, МИ, ПД}), {m, –m, m, –m**} ({ЗЕ, КФ, Р–, Р+}) and {cm, –cm, cm, –cm**} ({КВ, АК, З+, З–}).
The distinctive feature of normal subgroups, as already said, is the coincidence of the right and left cosets. Precisely for this reason, in all 4×4 blocks of Table 3 only four distinct IR operators each can be found. These are the same fours of operators that form the cosets of the normal subgroup we have chosen.
Table 3. Intertype relations table with simplified structure (in two operator-notation variants).
Formalized notation:
| y=Ax → | ILE | SEE | IEE | SLE | ILI | SEI | IEI | SLI | LII | ESI | LSI | EII | LIE | ESE | LSE | EIE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ILE | 1 | -1 | 1* | -1* | c | -c | c* | -c* | m | -m | -m* | m* | cm | -cm | -cm* | cm* |
| SEE | -1 | 1 | -1* | 1* | -c | c | -c* | c* | -m | m | m* | -m* | -cm | cm | cm* | -cm* |
| IEE | 1* | -1* | 1 | -1 | c* | -c* | c | -c | m* | -m* | -m | m | cm* | -cm* | -cm | cm |
| SLE | -1* | 1* | -1 | 1 | -c* | c* | -c | c | -m* | m* | m | -m | -cm* | cm* | cm | -cm |
| ILI | c | -c | c* | -c* | 1 | -1 | 1* | -1* | cm | -cm | -cm* | cm* | m | -m | -m* | m* |
| SEI | -c | c | -c* | c* | -1 | 1 | -1* | 1* | -cm | cm | cm* | -cm* | -m | m | m* | -m* |
| IEI | c* | -c* | c | -c | 1* | -1* | 1 | -1 | cm* | -cm* | -cm | cm | m* | -m* | -m | m |
| SLI | -c* | c* | -c | c | -1* | 1* | -1 | 1 | -cm* | cm* | cm | -cm | -m* | m* | m | -m |
| LII | m | -m | -m* | m* | cm | -cm | -cm* | cm* | 1 | -1 | 1* | -1* | c | -c | c* | -c* |
| ESI | -m | m | m* | -m* | -cm | cm | cm* | -cm* | -1 | 1 | -1* | 1* | -c | c | -c* | c* |
| LSI | m* | -m* | -m | m | cm* | -cm* | -cm | cm | 1* | -1* | 1 | -1 | c* | -c* | c | -c |
| EII | -m* | m* | m | -m | -cm* | cm* | cm | -cm | -1* | 1* | -1 | 1 | -c* | c* | -c | c |
| LIE | cm | -cm | -cm* | cm* | m | -m | -m* | m* | c | -c | c* | -c* | 1 | -1 | 1* | -1* |
| ESE | -cm | cm | cm* | -cm* | -m | m | m* | -m* | -c | c | -c* | c* | -1 | 1 | -1* | 1* |
| LSE | cm* | -cm* | -cm | cm | m* | -m* | -m | m | c* | -c* | c | -c | 1* | -1* | 1 | -1 |
| EIE | -cm* | cm* | cm | -cm | -m* | m* | m | -m | -c* | c* | -c | c | -1* | 1* | -1 | 1 |
| З– | З+ | КВ | АК | Р+ | Р– | ЗЕ | КФ | ПД | МИ | ДУ | ПО | ДЕ | РО | СЭ | ТО |
“Old” notation:
| y=Ax → | ILE | SEE | IEE | SLE | ILI | SEI | IEI | SLI | LII | ESI | LSI | EII | LIE | ESE | LSE | EIE |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| ILE | ТО | СЭ | РО | ДЕ | ПО | ДУ | МИ | ПД | ЗЕ | КФ | Р+ | Р– | КВ | АК | З– | З+ |
| SEE | СЭ | ТО | ДЕ | РО | ДУ | ПО | ПД | МИ | КФ | ЗЕ | Р– | Р+ | АК | КВ | З+ | З– |
| IEE | РО | ДЕ | ТО | СЭ | МИ | ПД | ПО | ДУ | Р– | Р+ | КФ | ЗЕ | З+ | З– | АК | КВ |
| SLE | ДЕ | РО | СЭ | ТО | ПД | МИ | ДУ | ПО | Р+ | Р– | ЗЕ | КФ | З– | З+ | КВ | АК |
| ILI | ПО | ДУ | МИ | ПД | ТО | СЭ | РО | ДЕ | КВ | АК | З– | З+ | ЗЕ | КФ | Р+ | Р– |
| SEI | ДУ | ПО | ПД | МИ | СЭ | ТО | ДЕ | РО | АК | КВ | З+ | З– | КФ | ЗЕ | Р– | Р+ |
| IEI | МИ | ПД | ПО | ДУ | РО | ДЕ | ТО | СЭ | З+ | З– | АК | КВ | Р– | Р+ | КФ | ЗЕ |
| SLI | ПД | МИ | ДУ | ПО | ДЕ | РО | СЭ | ТО | З– | З+ | КВ | АК | Р+ | Р– | ЗЕ | КФ |
| LII | ЗЕ | КФ | Р+ | Р– | КВ | АК | З– | З+ | ТО | СЭ | РО | ДЕ | ПО | ДУ | МИ | ПД |
| ESI | КФ | ЗЕ | Р– | Р+ | АК | КВ | З+ | З– | СЭ | ТО | ДЕ | РО | ДУ | ПО | ПД | МИ |
| LSI | Р– | Р+ | КФ | ЗЕ | З+ | З– | АК | КВ | РО | ДЕ | ТО | СЭ | МИ | ПД | ПО | ДУ |
| EII | Р+ | Р– | ЗЕ | КФ | З– | З+ | КВ | АК | ДЕ | РО | СЭ | ТО | ПД | МИ | ДУ | ПО |
| LIE | КВ | АК | З– | З+ | ЗЕ | КФ | Р+ | Р– | ПО | ДУ | МИ | ПД | ТО | СЭ | РО | ДЕ |
| ESE | АК | КВ | З+ | З– | КФ | ЗЕ | Р– | Р+ | ДУ | ПО | ПД | МИ | СЭ | ТО | ДЕ | РО |
| LSE | З+ | З– | АК | КВ | Р– | Р+ | КФ | ЗЕ | МИ | ПД | ПО | ДУ | РО | ДЕ | ТО | СЭ |
| EIE | З– | З+ | КВ | АК | Р+ | Р– | ЗЕ | КФ | ПД | МИ | ДУ | ПО | ДЕ | РО | СЭ | ТО |
| -cm* | cm* | cm | -cm | -m* | m* | m | -m | -c* | c* | -c | c | -1* | 1* | -1 | 1 |
As we see, the resulting table has “block symmetry” about the center, like the Liashkiavichius–Churyumov table. But the varieties of 4×4 blocks here number only four, not six. In each block, as already said, there are only 4 distinct operators.
If one looks at the IR table with simplified structure whose “body” is filled with the “old” operator notation (see Table 3, lower variant), the noted features of our table can also be discerned. But when the formalized IR operator notation is used, the structure of the table simply leaps to the eye.
Let us note nonetheless that the transformation rules for the abbreviated TIM designations under the action of the IR operators are quite well captured in the formalized notation we have proposed. Therefore the need for an IR table as a reference falls away of its own accord. All the same, independently composing various versions of the IR table would be a very useful exercise for students of socionics.
Conclusion
In the present article we have considered some questions connected with the possible variability of the IR table’s structure when it is built on the principle of ordering the list of TIMs into four distinguished fours, in each of which one and the same set of three IR operators “works.” To every such division of the socion into four fours of TIMs there corresponds its own 4-element subgroup of the 16-element group of IR operators (to the three operators is added the unit element of the group — the identity operator).
Of the 15 existing subgroups of order 4, only 7 are normal (or invariant, or self-conjugate). Only the normal 4-element subgroups generate a division of the socion into four fours of TIMs such that the IRs between representatives of any two fours can be specified by means of four operators.
The division of the socion into quadras does not correspond to a normal subgroup of IR operators. Therefore all possible IR tables based on the quadra-wise division of the socion also contain 4×4 blocks in each of which 8 distinct IR operators can be found.
The article gives a comparative analysis of three published IR tables based on the quadra-wise division of the socion, but with different orders of the right and left dual pairs in the last two quadras. It is shown that these seemingly insignificant changes in the order of TIMs exert a substantial influence on the structuredness of the IR table.
A new variant of the IR table has been proposed, connected with a division of the socion by means of the normal subgroup “Socionic temperament.” The new table has a simple structure which is especially well seen when the formalized IR operator notation proposed by us earlier is used.
Attention is drawn to the fact that the introduction of formalized notation for the IR operators makes it easy to do without memorizing any IR tables at all.
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The article was received by the editors on 29.12.2015.