A Mathematical Glossary of the Socion

Partition Spaces

• Monochotomy = the hexadecadic set which comprises the full Socion, or full relevant set, determined by context.
• Dichotomy = the division of a set into 2 distinct sets (traits, or octads in the context of the Socion).
• Tetrachotomy = the division of a set into 4 distinct sets, or tetrads in the case of the Socion (formed by the intersection of three nontrivial dichotomies).
• Octachotomy = the division of the Socion into 8 distinct sets, or dyads in the context of the Socion (formed by the intersection of 7 nontrivial dichotomies).
• Hexadecachotomy = the division of the Socion into its 16 monadic elements, or types of information metabolism (TIMs), and is formed by the intersection of all elements from a dichotomy system.

Mathematical Objects

Dichotomy Spaces and Traits

• $\mathfrak{d}$ = a dichotomy from a valid system of dichotomies where ${\mathfrak{d}}_0$ and ${\mathfrak{d}}_1$ are complementary traits. Alternatively, for any dichotomy, $D$ is used.
• $\mathfrak{D}$ = a system of dichotomies.
• $\mathcal{D}$ = the family of all orbital-respecting dichotomies (16 in total), with the subscript $i$ denoting the system. The list of all “orbital-respecting dichotomy systems” proceeds as follows…
  • ${\mathcal{D}}_1$ = Receptive-Adaptive Result Compass Process HEF
  • ${\mathcal{D}}_2$ = Semidual Mirage HEF
  • ${\mathcal{D}}_3$ = Kindred Business HEF
  • ${\mathcal{D}}_4$ = Flexible-Manoeuvring Result Compass Process HEF
  • ${\mathcal{D}}_5$ = Parallel Club Quadra Charged Rationality
  • ${\mathcal{D}}_6$ = Parallel Club Quadra Charged Rationality
  • ${\mathcal{D}}_7$ = Balanced-Stable Process Compass Result HEF
  • ${\mathcal{D}}_8$ = Perpendicular Club Quadra Vertedness
  • ${\mathcal{D}}_9$ = Activation Quasi-Identity HEF
  • ${\mathcal{D}}_{10}$ = Balanced-Stable Result Compass Process HEF
  • ${\mathcal{D}}_{11}$ = Linear-Assertive Result Compass Process HEF
  • ${\mathcal{D}}_{12}$ = Mirror Conflict HEF
  • ${\mathcal{D}}_{13}$ = Receptive-Adaptive Process Compass Result HEF
  • ${\mathcal{D}}_{14}$ = Tencer-Minaev
  • ${\mathcal{D}}_{15}$ = Reinin
  • ${\mathcal{D}}_{16}$ = Flexible-Manoeuvring Process Compass Result HEF
• $\mathbf{V}$ is the vector space associated with an $i$ dichotomy system in $\mathcal{D}$ that acts on the sociotypes and generate induced intertype relation groups, denoted ${\mathbf{V}}_i$. The vectors themselves are encoded via XOR logic, so for example the zero vector corresponds to the identity element. Sometimes XNOR logic is used to express the elements, where the identity element is represented by the one vector, but the former is mostly used. Refer to TIM Dichotomy Index for the basis vectors for each dichotomy system (these correspond to the 1st order dichotomies used for boolean algebras for a system of dichotomies).
• $\mathbb{D}$ denotes the universal collection of all of dichotomy systems $\mathfrak{D}$ that may act on the set of types, $T$.
• ${\mathcal{C}}_H$ is the class of dichotomy systems that preserve a subgroup $H<\mathbb{S}$. See TIM Octads Archive for more information.

Categories of Dichotomies

Mathematical Categories

• Orbital $\mathcal{O}$: The 7 Orbital dichotomies are the dichotomies that, when intersected, define the superego dyad and correspond bijectively to the index-2 subgroups of $\mathbb{S}$. For any two orbital dichotomies and a binary operation between them, the output will be another orbital dichotomy in the orbital subspace.
• Wall $\mathcal{W}$: These proport to the 8 non-orbital dichotomies in each system of dichotomies, with the complete family of wall dichotomies consisting of 144 total dichotomies. Each wall subspace is denoted $W_i$ which consists of 8 wall dichotomies associated with an $i$ dichotomy system. Every dichotomy space ${\mathcal{D}}_i$ consists of a wall subspace $W_i$ together with the 7 orbital dichotomies, not counting the tautological dichotomy. For any two wall dichotomies in $W_i$ and a binary operation between them, the output will be an orbital dichotomy. However, for the binary operation between a wall dichotomy and an orbital dichotomy will result in another wall dichotomy in the same wall subspace.
• Waffle: For a pair of dichotomy systems ${\mathcal{D}}_i$ and ${\mathcal{D}}_j$, and a pair of their associated wall subspaces $W_i$ and $W_j$, a “waffle” dichotomy is the output of the product of one dichotomy in $W_i$ and another in $W_j$. Exhausting this for all wall dichotomies in the relevant systems forms a wall subspace, that when combined with the 7 orbital dichotomies forms its associated independent dichotomy system. For any two waffle dichotomies in the same subspace and a binary operation between them, the output will always be an orbital dichotomy. However, for the binary operation between a waffle dichotomy and an orbital dichotomy will result in another waffle dichotomy in the same waffle subspace. Group-theoretic counting (via Lagrange’s Theorem and closure constraints) shows there are 15 independent waffle subspaces, with each subspace consisting of 8 waffle dichotomies. Moreover, all waffle dichotomies, like the orbital dichotomies, keep superego dyads invariant, meaning that for any trait in a waffle dichotomy $D$, with $D_0$ and $D_1$ being complementary traits in $D$, if a type $t$ is in the trait $D_0$ its superego counterpart $g(t)$ is also in $D_0$.

Dichotomy Classification Notation

Orbital Subsets

• The ‘Vector’ set $V$ is the set of dichotomies, that when intersected, form the ‘Challenge Response Groups’ tetrachotomy. The remaining 4 orbital dichotomies together form the ‘Converse’ set, denoted $C$.
• The ‘General’ set $G$ is the set of dichotomies, that when intersected, form the ‘Temperaments’ tetrachotomy. The remaining 4 orbital dichotomies together form the ‘Supralocal’ set, denoted $U$.
• The ‘Orientation’ set $O$ is the set of dichotomies, that when intersected, form the ‘Positivity Groups’ or ‘Forms of Will’ tetrachotomy. The remaining 4 orbital dichotomies together form the ‘Pivotal’ set, denoted $P$.
• The ‘Displacement’ set $\Delta$ is the set of dichotomies, that when intersected, form the ‘Displacement’ tetrachotomy. The remaining 4 orbital dichotomies form the ‘Extrapolative’ set, denoted $E$.
• The ‘Central’ set $Z$ is the set of dichotomies, that when intersected, form the ‘Stress Resistance’ tetrachotomy. The remaining 4 orbital dichotomies form the ‘Dihedral’ or ‘Square’ set, denoted $S$.

Wall Subsets (Tencer-Minaev and Reinin)

• The wall subspace $W_{14}$ can be divided into two sets, namely ‘Aristocratic’ and ‘Democratic’, denoted $A$ and $D$, respectively. The elements of $W_{14}$ are as follows:
  • $A_1$ = 1st-Internal / 1st-External
  • $A_2$ = 1st-Delta / 1st-Beta
  • $A_3$ = 2nd-External / 2nd-Internal
  • $A_4$ = 2nd-Beta / 2nd-Delta
  • $D_1$ = 1st-Abstract / 1st-Involved
  • $D_2$ = 1st-Alpha / 1st-Gamma
  • $D_3$ = 2nd-Abstract / 2nd-Involved
  • $D_4$ = 2nd-Alpha / 2nd-Gamma
• The wall subspace $W_{15}$ can be divided into two sets, namely ‘Rational’ and ‘Irrational’, denoted $R$ and $I$, respectively. The elements of $W_{15}$ are as follows:
  • $R_1$ = Logical / Ethical
  • $R_2$ = Merry / Serious
  • $R_3$ = Constructivist / Emotivist
  • $R_4$ = Yielding / Obstinate
  • $I_1$ = Intuitive / Sensory
  • $I_2$ = Judicious / Decisive
  • $I_3$ = Tactical / Strategic
  • $I_4$ = Carefree / Farsighted

Intertype Relations

• $\mathbb{S}$: This is the group of all intertype relations in Socionics. This group is isomorphic to $D_4\times {\mathbb{Z}}_2$ (the direct product of the dihedral group of order 8 and the cyclic group of order 2). The elements of $\mathbb{S}$ are as follows:
  • $e$ identity, or identical
  • $d$ dual
  • $a$ activator
  • $m$ mirror
  • $g$ superego
  • $c$ conflict
  • $q$ quasi-identical
  • $x$ extinguishment, or contrary
  • $S$ supervisor
  • $B$ benefactor
  • $k$ kindred
  • $h$ semidual, or half-dual
  • $s$ supervisee
  • $b$ beneficiary
  • $l$ business, or lookalike
  • $i$ mirage, or illusory