Abelian kernels of some monoids of injective partial transformations and an application
The purpose of the paper is to calculate the Abelian kernels of the monoids \(\text{POI}_n\) and \(\text{POPI}_n\). The calculation relies on presentations for the monoids \(\text{POI}_n\) and \(\text{POPI}_n\) devised by the second author which appear elsewhere. In particular from the results we can infer that the Abelian kernel of \(\text{POPI}_4\) is not aperiodic so that the pseudovariety \(\mathbf{POPI}\) generated by the monoids \(\text{POPI}_n\) of all partial injective and orientation-preserving mappings on finite chains is not contained in the Malcev product of \(\mathbf{POI}\), the pseudovariety generated by the monoids of all partial injective and order-preserving mappings on finite chains, with \(\mathbf{Ab}\), the pseudovariety of all Abelian groups. More specifically we then have \(\mathbf{POPI}\nsubseteq\mathbf{POI}*\mathbf{Ab}\). Whether or not the negation of the reverse inclusion \(\mathbf{POI}*\mathbf{Ab}\nsubseteq\mathbf{POPI}\) is true remains unresolved but the left-hand side does lie inside the pseudovariety of all monoids with Abelian subgroups as the same is true of the Malcev product of all aperiodic monoids with Abelian groups.
scientific article; zbMATH DE number 1545255