Ends of Semigroups

TL;DR

This paper introduces a new concept of the ends of a semigroup's Cayley graph, proving invariance properties and an analogue of Hopf's theorem for left cancellative semigroups, extending classical group results.

Contribution

It defines the ends of semigroups via Cayley graphs and establishes their invariance under generating set changes and subsemigroup extensions, along with a Hopf-type theorem for semigroups.

Findings

Ends of semigroups are invariant under generating set changes.

The number of ends is preserved in subsemigroups and extensions of finite Green index.

Left cancellative semigroups have 1, 2, or infinitely many ends.

Abstract

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.