Representations of the inverse hull of a 0-left cancellative semigroup

A semigroup S containing a zero element is said to be 0-left cancellative if st = sr neq 0 implies that t = r. Given such an S we build an inverse semigroup H(S), called the inverse hull of S. Motivated by the study of certain C*-algebras associated to H(S) (a task that we will address in a subsequent article) we carry out a detailed analysis of the spectrum of the idempotent semilattice E(S) of H(S) with a special interest in identifying the ultra-characters. In order to produce examples of characters on E(S), we introduce the notion of strings in a semigroup, attempting to make sense of the infinite paths which are fundamental in the study of graph C*-algebras. Our strongest results are obtained under the assumption that S admits least common multiples, but we also touch upon the notion of finite alignment, motivated by the corresponding notion from the theory of higher rank graphs, and which has also appeared in recent papers by Spielberg and collaborators.