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 .