Order theoretic variants of the fundamental theorem of compact semigroups
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 remarkably, no single element in \(\uparrow M\) can lie above two different elements of \(M\), a conclusion which leads to a semigroup congruence on \(\uparrow M\). There is a somewhat weaker analogue for \(\downarrow M\).
scientific article; zbMATH DE number 638354