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Some aspects of Green’s relations on periodic semigroups. (English) Zbl 0538.20030
A semigroup S is called periodic if to each element a of S there corresponds an idempotent e and a positive integer n such that \(a^ n=e\). Let S be a periodic semigroup, and \({\mathcal K}\) the equivalence relation on S defined by a\({\mathcal K}b\) if and only if \(a^ m=b^ n=e\) for some integers m, n and an idempotent e. The main purpose of this paper is to give a characterization of those periodic semigroups for which Green’s relation \({\mathcal J}\) is included in \({\mathcal K}\). In particular, the following results are established: (A) For a periodic semigroup S the following (1)-(3) are equivalent. (1) S is a semilattice of unipotent semigroups; (2) S is a union of unipotent semigroups and is weakly commutative; (3) \({\mathcal J}\subseteq {\mathcal K}\). (B) The following (1)-(3) are equivalent for a periodic combinatorial semigroup S. (1) S is a semilattice of nil semigroups; (2) S is a union of nil semigroups and is weakly commutative; (3) S is \({\mathcal J}\)-trivial.
Reviewer: M.Yamada

MSC:
20M10 General structure theory for semigroups
20M15 Mappings of semigroups
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