On linear operators strongly preserving invariants of Boolean matrices

Yizhi Chen; Xian Zhong Zhao

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 1, page 169-186
  • ISSN: 0011-4642

Abstract

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Let 𝔹 k be the general Boolean algebra and T a linear operator on M m , n ( 𝔹 k ) . If for any A in M m , n ( 𝔹 k ) ( M n ( 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 𝔹 k . Meanwhile, noting that a general Boolean algebra 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 𝔹 k from another point of view.

How to cite

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Chen, Yizhi, and Zhao, Xian Zhong. "On linear operators strongly preserving invariants of Boolean matrices." Czechoslovak Mathematical Journal 62.1 (2012): 169-186. <http://eudml.org/doc/247108>.

@article{Chen2012,
abstract = {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.},
author = {Chen, Yizhi, Zhao, Xian Zhong},
journal = {Czechoslovak Mathematical Journal},
keywords = {linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra; linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra; linear preserver problem},
language = {eng},
number = {1},
pages = {169-186},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On linear operators strongly preserving invariants of Boolean matrices},
url = {http://eudml.org/doc/247108},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Chen, Yizhi
AU - Zhao, Xian Zhong
TI - On linear operators strongly preserving invariants of Boolean matrices
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 169
EP - 186
AB - 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.
LA - eng
KW - linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra; linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra; linear preserver problem
UR - http://eudml.org/doc/247108
ER -

References

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