Linear maps that strongly preserve regular matrices over the Boolean algebra

Abstract

The set of all m × n Boolean matrices is denoted by $$ \mathbb{M} $$ m,n . We call a matrix A ∈ $$ \mathbb{M} $$ m,n regular if there is a matrix G ∈ $$ \mathbb{M} $$ n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $$ \mathbb{M} $$ m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $$ \mathbb{M} $$ m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $$ \mathbb{M} $$ m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $$ \mathbb{M} $$ m,n , or m = n and T(X) = UX T V for all X ∈ $$ \mathbb{M} $$ n .