Abelian kernels of some monoids of injective partial transformations and an application (Q1590072)

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.