On linear operators strongly preserving invariants of Boolean matrices

Abstract

Let U k be the general Boolean algebra and T a linear operator on M m,n (U k ). If for any A in M m,n (U k ) (M n (U k ), respectively), A is regular (invertible, respectively) if and only if T(A) is regular (invertible, respectively), then T is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over U k . Meanwhile, noting that a general Boolean algebra U k is isomorphic to a finite direct product of binary Boolean algebras, we also give some characterizations of linear operators that strongly preserve regular (invertible, respectively) matrices over 169-7 k from another point of view.