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Linear operators strongly preserving idempotent matrices over semirings. (English) Zbl 0744.15010
Let $S$ be a semiring, let $M\sb n(S)$ be the set of $n\times n$ matrices over $S$. A linear operator $T$ on $M\sb n(S)$ is said to be strongly $r$-potent preserving if $T(X)\sp r=T(X)$ if and only if $X\sp r=X$. These operators form a semigroup $\cal S$. In this paper we have $r=2$. If $S=\{0,1\}$ is the 2-element Boolean algebra then $\cal S$ is generated by the transposition and the similarity operators $X\mapsto PXP\sp t$, $P$ a permutation matrix. If $S$ is antinegative (i.e. no element $\ne 0$ has an additive inverse) and free of zero divisors then $\cal S$ is generated by the transposition, the similarity operators $X\mapsto AXA\sp{-1}$ and those operators $T\sb A(X)=A\circ X$ that are contained in $\cal S$ ($\circ$ denotes the Hadamard product). If moreover each element of $S$ is idempotent then such operators $T\sb A$ occur only for $n=2$.

15B33Matrices over special rings (quaternions, finite fields, etc.)
15A60Applications of functional analysis to matrix theory
Full Text: DOI
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