The inverse hull of 0-left cancellative semigroups

TL;DR

This paper introduces the inverse hull of zero-left cancellative semigroups, explores its structure when least common multiples exist, and relates it to C*-algebras in symbolic dynamics.

Contribution

It constructs the inverse hull for zero-left cancellative semigroups and analyzes its idempotent semilattice and spectrum, connecting it to C*-algebras from symbolic dynamics.

Findings

Constructed the inverse hull H(S) for zero-left cancellative semigroups.

Analyzed the spectrum of the idempotent semilattice of H(S).

Connected H(S) to C*-algebras associated with subshifts.

Abstract

Given a semigroup S with zero, which is left-cancellative in the sense that st=sr \neq 0 implies that t=r, we construct an inverse semigroup called the inverse hull of S, denoted H(S). When S admits least common multiples, in a precise sense defined below, we study the idempotent semilattice of H(S), with a focus on its spectrum. When S arises as the language semigroup for a subsift X on a finite alphabet, we discuss the relationship between H(S) and several C*-algebras associated to X appearing in the literature.