On the relative solvability of certain inverse monoids. (Q617855)

If \(\mathbf H\) is a group pseudovariety and \(M\) is a finite monoid, then the \(\mathbf H\)-kernel of \(M\) is the set of all elements of \(M\) related to 1 under any relational morphism from \(M\) into a group from \(\mathbf H\). It is a submonoid of \(M\) containing all idempotents of \(M\). The construction can be iterated, and \(M\) is said to be \(\mathbf H\)-solvable if iterating the computation of the \(\mathbf H\)-kernel one eventually arrives at the submonoid generated by the idempotents of \(M\). This concept was suggested by \textit{M. Delgado} and \textit{V. H. Fernandes} [Int. J. Algebra 15, No. 3, 547-570 (2005; Zbl 1083.20047)] who also proved that an inverse monoid is \(\mathbf H\)-solvable if and only if each of its maximal subgroups is \(\mathbf H\)-solvable. The authors and \textit{V. H. Fernandes} [Bull. Aust. Math. Soc. 73, No. 3, 375-404 (2006; Zbl 1104.20053)] studied the \(\mathbf{Ab}\)-solvability of four series of inverse monoids where \(\mathbf{Ab}\) stands for the pseudovariety of all finite Abelian groups. In the present paper, the authors study the \(\mathbf H\)-solvability of the same monoids with \(\mathbf H\) being an arbitrary subpseudovariety of \(\mathbf{Ab}\). The monoids investigated consist of partial injective transformations of a finite chain that are 1) order-preserving, 2) order-preserving or order-reversing, 3) orientation-preserving, or 4) orientation-preserving or orientation-reversing.