Order theoretic variants of the fundamental theorem of compact semigroups
type: Item
instance of: scholarly article
title: Order theoretic variants of the fundamental theorem of compact semigroups
zbMATH Open document ID: 0812.22003
DOI: 10.1007/BF02573485
author: S. D. Hippisley-Cox
published in: Semigroup Forum
publication date: 1994-10-12T00:00:00Z
full work available at URL: https://eudml.org/doc/135348
review text: This paper offers an extensive study of po-semigroups (partially ordered semigroups with order-preserving multiplications) with particular emphasis on the fundamental structure theorem -- that is, the possession of a minimal ideal which is a paragroup. The existence of minimal ideals if often deduced from compactness conditions (for example, they are always found in compact right topological semigroups), and the author is aiming at replacing these by order conditions. He observes that order properties are left-right symmetric, whereas right continuity is not. The key to the structure theorem is the existence of an idempotent minimal in the algebraic sense. The author observes that in each of the theories which involve partially ordered semigroups, each semigroup contains an idempotent of this type which is also order maximal in the minimal left ideal which contains it. General conditions for the existence of such idempotents are given. Special cases are that they do if (a) there are a finite number of maximal, and of minimal, elements in the po-semigroup, and if (b) there are a finite number of minimal elements and the ascending chain condition is satisfied. The more striking part of the paper is concerned with the structural consequences of the existence of such an idempotent in a po-semigroup \(S\). For example, the maximal subgroups of the minimal ideal must all be antichains. The basic paragroup structure of the minimal ideal is an order isomorphism too (with the direct product having the product order). The set of maximal elements of any minimal left ideal is itself a left group. The subsets \(L\) of \(S\) which are minimal with respect to the property of being both a left ideal and a ``down-set'' (\(L = \downarrow L = \{x : x \leq y\) for some \(y \in L\}\)) are either identical or disjoint, and each such \(L\) contains precisely one minimal left ideal. If \(M\) denotes the set of elements order maximal in \(L\), then \((\uparrow M)(\downarrow M) = M\). Quite …
Mathematics Subject Classification ID: 22A15
Mathematics Subject Classification ID: 06F05
Mathematics Subject Classification ID: 20M12
zbMATH DE Number: 638354
zbMATH Keywords: partially ordered semigroups
zbMATH Keywords: po-semigroups
zbMATH Keywords: minimal ideals
zbMATH Keywords: compactness conditions
zbMATH Keywords: compact right topological semigroups
zbMATH Keywords: idempotents
zbMATH Keywords: minimal elements
zbMATH Keywords: maximal subgroups
zbMATH Keywords: antichains
zbMATH Keywords: maximal elements
zbMATH Keywords: semigroup congruence
reviewed by: J. Pym
MaRDI profile type: Publication
cites work: Q3995290
cites work: Q3848243
cites work: Q4692885
cites work: Compact right uppersemicontinuous semigroups of closed sets
cites work: On the structure of certain po-semigroups
cites work: On Bounded po-Semigroups
OpenAlex ID: W2073299005
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description: scientific article; zbMATH DE number 638354