A class of inverse monoids acting on ordered forests

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.