Groebner-Shirshov bases for free inverse semigroups

TL;DR

This paper constructs a Groebner-Shirshov basis for free inverse semigroups, providing a systematic method to find unique normal forms of words and an algorithm for normalization.

Contribution

It introduces a Groebner-Shirshov basis for free inverse semigroups based on previous constructions, enabling effective normalization of words.

Findings

Established a Groebner-Shirshov basis for free inverse semigroups

Provided an algorithm to convert words into their normal forms

Identified the shortest unique normal forms for equivalence classes

Abstract

A new construction of a free inverse semigroup was obtained by Poliakova and Schein in 2005. Based on their result, we find a Groebner-Shirshov basis of a free inverse semigroup relative to the deg-lex order of words. In particular, we give the (unique and shortest) Groebner-Shirshov normal forms in the classes of equivalent words of a free inverse semigroup together with the Groebner-Shirshov algorithm to transform any word to its normal form.