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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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