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Linear operators strongly preserving idempotent matrices over semirings. (English) Zbl 0744.15010

Let S be a semiring, let M n (S) be the set of n×n matrices over S. A linear operator T on M n (S) is said to be strongly r-potent preserving if T(X) r =T(X) if and only if X r =X. These operators form a semigroup 𝒮. In this paper we have r=2.

If S={0,1} is the 2-element Boolean algebra then 𝒮 is generated by the transposition and the similarity operators XPXP t , P a permutation matrix.

If S is antinegative (i.e. no element 0 has an additive inverse) and free of zero divisors then 𝒮 is generated by the transposition, the similarity operators XAXA -1 and those operators T A (X)=AX that are contained in 𝒮 ( denotes the Hadamard product). If moreover each element of S is idempotent then such operators T A occur only for n=2.


MSC:
15A33Matrices over special rings
15A60Applications of functional analysis to matrix theory