On linear operators strongly preserving invariants of Boolean matrices

Chen, Yizhi ; Zhao, Xianzhong

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 1, pp. 169-186.

Résumé

Let ${\mathbb{B}}_k$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}({\mathbb{B}}_k)$. If for any $A$ in $M_{m,n}({\mathbb{B}}_k)$ ($M_n({\mathbb{B}}_k)$, 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 ${\mathbb{B}}_k$. Meanwhile, noting that a general Boolean algebra ${\mathbb{B}}_k$ 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 ${\mathbb{B}}_k$ from another point of view.

Détail

DOI : 10.1007/s10587-012-0004-y

Classification : 06E05, 15A04, 15A09, 15A86, 15B34, 16Y60
Mots-clés : linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra