Michael John

This paper investigates the enumeration of elements in three classes of signed partial transformation semigroups, namely the partial signed order-preserving semigroup (PSPO_n), the partial signed order-decreasing semigroup (PSPD_n), and the partial signed order-preserving-or-order-reversing semigroup (PSPOD_n). Previous computations provide the initial cardinality sequences [ |PSPO_n| = 1, 13, 133, 1281, \ldots, ] [ |PSPD_n| = 1, 9, 81, 819, \ldots, ] and [ |PSPOD_n| = 1, 16, 209, 2302, \ldots, ] for (1 \leq n \leq 4). A numerical inconsistency in the reported data for (PSPD_4) is examined: the component values (191) and (700) sum to (891), whereas the reported cardinality is (819). Consequently, if the stated total (819) is retained, the corresponding first component must be (119). To provide a systematic framework for extending these finite enumerations, we define the cardinality functions [ A_n = |PSPO_n|, \qquad B_n = |PSPD_n|, \qquad C_n = |PSPOD_n|, ] and introduce the polarity-based enumeration function [ \mathcal{P}_{S}(n) ================== N_{S}^{-}(n) + N_{S}^{\ast}(n), \qquad S \in {O,D,OD}. ] A refined domain-rank enumeration framework is proposed as [ \mathcal{P}_{S}(n) ================== \sum_{k=1}^{n} \binom{n}{k} \sum_{r=1}^{k} E_{S}(n,k,r), ] where (E_{S}(n,k,r)) represents the number of admissible signed partial transformations of domain size (k) and rank (r) satisfying the structural condition (S). Using Newton finite-difference interpolation on the known cardinalities, computational formulae are obtained as [ \widehat{A}_n ============= 1 * 12\binom{n-1}{1} * 108\binom{n-1}{2} * 920\binom{n-1}{3}, ] [ \widehat{B}_n ============= 1 * 8\binom{n-1}{1} * 64\binom{n-1}{2} * 602\binom{n-1}{3}, ] and [ \widehat{C}_n ============= 1 * 15\binom{n-1}{1} * 178\binom{n-1}{2} * 1722\binom{n-1}{3}. ] These expressions reproduce exactly the available cardinalities for (1 \leq n \leq 4) and provide a systematic procedure for generating conjectural values for larger (n). The interpolated expressions are distinguished from exact semigroup enumeration formulae and are therefore treated as computational conjectures pending verification from the defining transformation conditions. The resulting framework provides a basis for computational enumeration, recurrence analysis, conjecture testing, and the subsequent derivation of exact closed-form counting formulae for these finite signed partial transformation semigroups.

The interaction between finite algebraic structures and cryptographic systems motivates the search for unified frameworks that can connect homomorphism enumeration, quotient constructions, fuzzy algebraic actions, and conjugacy-class methods. In this paper, a structural framework is developed for finite groups acting on near-rings and for homomorphic images that induce modular B-algebras. The paper is motivated by previous works of the author on B-algebras generated by modulo integer groups, conjugacy-class key agreement, fuzzy group actions on near-rings, and homomorphism enumeration from the quaternion group. We define homomorphic complexity indices, fuzzy stabilizer indices, conjugacy complexity indices, and induced modular B-algebra invariants. Several results are proved: kernels of homomorphisms act trivially under induced actions; cyclic homomorphic images give canonical B-algebras; fuzzy stabilizers are subgroups; orbit-stabilizer relations persist in the fuzzy-invariant setting; and conjugacy-based key agreement can be interpreted through commuting subgroups and algebraic invariants. Examples involving cyclic groups, the quaternion group, dihedral groups, and nilpotent groups illustrate the theory. The framework gives a mathematical basis for combining quotient-based algebra, fuzzy symmetry, and conjugacy structures in the analysis of algebraic cryptographic protocols.