The Automorphism Group of Idempotent Matrices Preservers over the Boolean Algebra

Abstract

In this paper, we consider the groups of linear operators that preserve idempotent matrices over the binary Boolean algebra. We prove that this set forms an automorphism group under composition and determine its exact algebraic structure as a direct product Sn×C2 for n≥2. Furthermore, we distinguish between elementary transposition matrices and involutory permutation matrices. We establish a recurrence relation, a closed combinatorial formula and asymptotic estimates for the number of involutory permutation matrices, showing that these numbers coincide with the involution numbers in the symmetric group. Structural properties such as center, solvability and generating sets of this group are fully characterized.