A class of inverse monoids acting on ordered forests

Abstract

Inverse monoid actions on ordered forests are studied to generalize the Bass-Serre theory to a certain class of inverse monoids. We introduce the concepts of graphs of inverse monoids and their fundamental inverse monoids and discuss their basic properties. Using these concepts, we characterize the inverse monoids acting on ordered forests satisfying some conditions as the groups acting on trees without inversion are characterized as the fundamental groups of graphs of groups. We also investigate the local action of a maximal subgroup of the fundamental inverse monoid on a connected component of the universal cover and obtain a presentation of the maximal subgroup.  2004 Elsevier Inc. All rights reserved.

A topological approach to inverse and regular semigroups

“…While this will not be important in our current work, we do mention that it is connected with the notion of an ordered forest as per [25].…”

“…We mention that Yamamura [25] defines a notion of a graph of full inverse monoids and the fundamental inverse monoid of such. Full amalgams and full HNN-extensions are special cases of this construction.…”

“…We conjecture that actions of inverse semigroups on ordered forests, as per [25], can be completely classified using ordered 2-complexes and the usual translation between Bass-Serre theory and graphs and trees of 2-complexes. In particular, we guess the solution will obtain the structure of such as the fundamental ordered groupoid of an ordered graph of groups.…”

Ends for subsemigroups of finite index

“…The reason that we did not further pursue this idea is the fact that the cancellative semigroup (N, +) acts on the ray-tree N in the natural way with finite edge stabilizers and without globally fixed points, but still has 1 end. Developing Bass-Serre theory for cancellative semigroups, akin to such theory for inverse semigroups created by Yamamura [24] might be a useful approach.…”

Embedding theorems for HNN extensions of inverse semigroups

“…An HNN extension (2.1) is called full if E(A) = E(B) = E(S). A full HNN extension can be characterized as a fundamental inverse monoid of a loop of inverse monoids, and this is employed to study the class of inverse monoids acting on ordered forests in [19]. This is considered as a generalization of the Bass-Serre theory.…”