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A note on indecomposable elements in the tensor product of semigroups. (English) Zbl 0584.20052
An element s of a semigroup S is said to be indecomposable if \(s\in S\setminus S^ 2\). In this paper the influence of indecomposable elements of semigroups A, B on the structure of the tensor product \(A\otimes B\) is studied. If \(| A\setminus A^ 2| >1\) and \(| B\setminus B^ 2| >1\), then \(A\otimes B\) is infinite and non- commutative. For \(A\otimes B\) to be finite, at least one of \(| A\setminus A^ 2|\) and \(| B\setminus B^ 2|\) has to be equal to 1. The semigroups \((A/A^ 2)\otimes (B/B^ 2)\) and \(A\otimes B/(A\otimes B)^ 2\) are isomorphic if and only if \(| A\setminus A^ 2| \leq 1\) or \(| B\setminus B^ 2| \leq 1\). The tensor product \(A\otimes B\) is globally idempotent if and only if A or B is globally idempotent.
Reviewer: H.Jürgensen

MSC:
20M10 General structure theory for semigroups
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References:
[1] GRILLET P. A.: The tensor product of semigroups. Trans. Amer. Math. Soc. 138, 1969, 267-280. · Zbl 0191.01601
[2] GALANOVÁ J.: Codomain of the tensor product of semigroups. Math. Slovaca · Zbl 0584.20051
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