A remark on contracting inverse semigroups

TL;DR

This paper investigates inverse semigroups with tree-like idempotent semi-lattices, providing a necessary and sufficient condition for the semigroup to be contracting, aligning more with contracting actions than previous criteria.

Contribution

It introduces a new characterization of contracting inverse semigroups with tree-like idempotent semi-lattices, refining earlier conditions by relating to contracting actions.

Findings

Characterization of contracting inverse semigroups with tree-like semi-lattices

Condition that all tight filters are ultra-filters

New perspective closer to contracting actions

Abstract

A semi-lattice is said to be tree-like when any two of its elements are either orthogonal or comparable. Given an inverse semigroup S whose idempotent semi-lattice is tree-like, and such that all tight filters are ultra-filters, we present a necessary and sufficient condition for S to be contracting which looks closer in spirit to the notion of contracting actions than a recent condition found by the second named author and E. Pardo.