Monoids respecting \(n\)-chains of intervals
Let \(\mathbb{O}\) (resp., \(\mathbb{P}\mathbb{O}\)) denote the monoid pseudovariety generated by all monoids of (partial) order-preserving mappings on a finite chain. Starting with these pseudovarieties, the authors construct two strict hierarchies \(\mathbb{O}\subset\mathbb{O}^2\subset\ldots\subset\mathbb{O}^n\subset\ldots\) and \(\mathbb{P}\mathbb{O}\subset\mathbb{P}\mathbb{O}^2\subset\ldots\subset\mathbb{P}\mathbb{O}^n\subset\ldots\) of aperiodic monoid pseudovarieties both giving as the union the pseudovariety of all finite aperiodic monoids. The members of the hierarchies are defined in terms of their generating transformation monoids that respect a certain chain of interval partitions of a finite chain. On the other hand, the \(n\)th member of each hierarchy can be obtained as the \(n\)th semidirect power of the first member (Theorem 4). The second hierarchy can be obtained from the first one by semidirectly multiplying through on the left by the pseudovariety of all finite semilattices (Theorem 8). This implies that the hierarchies intermingle in the following way: \(\mathbb{O}\subset\mathbb{P}\mathbb{O}\subset\mathbb{O}^2\subseteq\mathbb{P}\mathbb{O}^2\subseteq\ldots\subseteq\mathbb{O}^n\subseteq\mathbb{P}\mathbb{O}^n\subseteq\mathbb{O}^{n+1}\subseteq\ldots\) (Corollary 9). The question as to whether \(\mathbb{O}^n\neq\mathbb{P}\mathbb{O}^n\) for all \(n\) remains open. The authors also find, for each \(n\), a pseudoidentity valid in \(\mathbb{O}^n\) (Proposition 10) but not in the pseudovariety \(\mathbb{L}\) of all finite \(\mathcal L\)-trivial semigroups (Proposition 12) thus proving that \(\mathbb{L}\) belongs to no finite level of any of the two hierarchies.
partial order-preserving mappings of finite chains
scientific article; zbMATH DE number 1014037