We introduce two generalisations of the full symmetric inverse semigroup IX and its dual semigroup I ∗ X – inverse semigroups PI ∗ X and PI ∗ X . Both of them have the same carrier and contain IX . Binary operations on PI∗X and PI ∗ X are reminiscent of the multiplication in IX . We use a convenient geometric way to realise elements from these two semigroups. This enables us to study efficientl...

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon’s approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix...

In this work, we show that a strongly continuous semigroup generated by a normal operator N is conjugate to the semigroup generated by the real part of N , provided zero is not an eigenvalue of the real part of N . We also show that in case N satisfies a certain sectorial property, the homeomorphism establishing the conjugacy, as well as its inverse, is locally Hölder continuous. Moreover, in c...

We prove that the Schützenberger graph of any element of the HNN-extension of a finite inverse semigroup S with respect to its standard presentation is a context-free graph in the sense of [11], showing that the language L recognized by this automaton is context-free. Finally we explicitly construct the grammar generating L, and from this fact we show that the word problem for an HNN-extension ...

We construct the inverse partition semigroup IPX , isomorphic to the dual symmetric inverse monoid I∗ X , introduced in [6]. We give a convenient geometric illustration for elements of IPX . We describe all maximal subsemigroups of IPX and find a generating set for IPX when X is finite. We prove that all the automorphisms of IPX are inner. We show how to embed the symmetric inverse semigroup in...

A subset U of a set S with a binary operation is called avoidable if S can be partitioned into two subsets A and B such that no element of U can be written as a product of two distinct elements of A or as the product of two distinct elements of B. The avoidable sets of the bicyclic inverse semigroup are classified.

Assume G is a group definable in a model M of a stable theory T . We prove that the semigroup SG(M) of complete G-types over M is an inverse limit of some semigroups type-definable in M eq. We prove that the maximal subgroups of SG(M) are inverse limits of some definable quotients of subgroups of G. We consider the powers of types in the semigroup SG(M) and prove that in a way every type in SG(...