Archimedean ordered semigroups as ideal extensions
The main result of the paper is a structure theorem concerning the ideal extensions of Archime\-dean ordered semigroups. Let \((S,\cdot,\leq )\) be an ordered semigroup. The authors prove that if \(S\) is Archimedean and \(e\) is an idempotent element of \(S\), then there exists an ideal \(K\) of \(S\) containing \(e\) such that \(K\) is a simple subsemigroup of \(S\) and the ordered semigroup \(S|K\) is nil. Conversely, let \(K\) be an ideal of \(S\) such that \(K\) is a simple subsemigroup of \(S\) and the ordered semigroup \(S|K\) be nil. Then \(S\) is Archimedean. By the above conclusions, the authors obtain the following results: Let \(S\) be an ordered semigroup. If \(S\) is Archimedean and contains an idempotent, then \(S\) is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup. Conversely, if \(S\) is an ideal extension of a simple ordered semigroup by a nil ordered semigroup, then \(S\) is Archimedean. Especially, an ordered semigroup \(S\) is Archimedean and contains an idempotent if and only if \(S\) is an ideal extension of a simple ordered semigroup containing an idempotent by a nil ordered semigroup.
ideal extension of an ordered semigroup by an ordered semigroup
scientific article; zbMATH DE number 5547495
scientific article