Article
MSC:
06E05,
15A04,
15A09,
15A86,
15B34,
16Y60 | MR 2899743 | Zbl 1249.15034 | DOI: 10.1007/s10587-012-0004-y
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Keywords:
linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra
linear operator; invariant; regular matrix; invertible matrix; general Boolean algebra
Summary:
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.
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