Normally ordered inverse semigroups
Let \(S\) be an inverse semigroup and \(E\) the set of its idempotents. Suppose that there exists a partial order \(\ll\) on \(E\) such that two idempotents are \(\ll\)-comparable if and only if they belong to the same \(\mathcal J\)-class of \(S\) and, for all \(s\in S\) and \(e,f\in Ess^{-1}\), \(e\ll f\) implies \(s^{-1}es\ll s^{-1}fs\). Then \(S\) is called a normally ordered inverse semigroup. Groups, semilattices, 0-simple inverse semigroups are normally ordered, as well as the semigroup \(\mathcal{POI}_n\) of all partial order preserving transformations of the \(n\)-element chain (Proposition 2.1). A finite inverse semigroup \(S\) is normally ordered if and only if its quotient over the maximum idempotent separating congruence is normally ordered (Corollary 2.7). A finite normally ordered inverse semigroup \(S\) is fundamental if and only if \(S\) is aperiodic and if and only if \(S\) embeds into the semigroup \(\mathcal{POI}_n\) where \(n=| E|\) (Theorem 3.2 and Corollary 3.3). The class \(\mathbf{NO}\) of all finite normally ordered inverse semigroups forms a pseudovariety of inverse semigroups (Theorem 2.5). This pseudovariety is closely related with the pseudovariety \(\mathbf{PCS}\) generated by all semigroups \(\mathcal{POI}_n\), which was studied by \textit{D. F. Cowan} and \textit{N. R. Reilly} [Int. J. Algebra Comput. 5, No. 3, 259-287 (1995; Zbl 0834.20063)]. Namely, \(\mathbf{PCS}\) coincides with the class of aperiodic semigroups in \(\mathbf{NO}\) (Theorem 3.4); on the other hand, \(\mathbf{NO}\) is shown to be generated by \(\mathbf{PCS}\) together with the class of all finite groups (Theorem 5.2). The author also finds a class of generators for the pseudovariety \(\mathbf{NO}\) (Theorem 4.4) and shows that \(\mathbf{NO}\) has no finite pseudoidentity basis (Corollary 3.8).
order preserving transformations of chains
scientific article; zbMATH DE number 1156731