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Semigroups whose proper one-sided ideals are t-Archimedean. (English) Zbl 0601.20055

This paper studies semigroups for which certain classes of subsemigroups satisfy special constraints. If S is a semigroup, let \(a(S)\) be a collection of subsemigroups of S, e.g., \(a(S)=r(S)\), the collection of right ideals of S, or \(a(S)=\ell(S)\), the collection of left ideals of S. Let P be some fixed collection of semigroups, e.g., \(P=REG\), the collection of regular semigroups, or \(P=RAS\), the collection of right archimedean semigroups. Let \(P/a=\{S:\) \(a(S)\subseteq P\}\) and \(P/\bar a= \{S:\) \(a(S)\setminus \{S\}\subseteq P\}.\)
This paper explores the relation between sets P and \(P/a\) or \(P/\bar a\) for various values of P and a. It culminates in the result that every proper one-sided ideal of a semigroup S is t-archimedean iff S satisfies one of the following conditions. 1. S is t-archimedean. 2. S is a semilattice of semigroups M and T, where M is a t-archimedean ideal of S and T a group whose identity is the identity of S. 3. S is a left (right) zero-semigroup of two groups.
Reviewer: W.R.Nico

MSC:

20M10 General structure theory for semigroups
20M12 Ideal theory for semigroups
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