Avoidable Sets in The Bicyclic Inverse Semigroup

TL;DR

This paper classifies avoidable sets within the bicyclic inverse semigroup, providing a comprehensive understanding of their structure and properties in this algebraic context.

Contribution

It introduces a classification of avoidable sets specifically in the bicyclic inverse semigroup, a novel focus in algebraic semigroup research.

Findings

Complete classification of avoidable sets in the bicyclic inverse semigroup

Identification of structural properties of avoidable sets

Insights into partitioning of semigroup elements

Abstract

A subset $U$ of a set $S$ with a binary operation is called {\it avoidable} if $S$ can be partitioned into two subsets $A$ and $B$ such that no element of $U$ can be written as a product of two distinct elements of $A$ or as the product of two distinct elements of $B$. The avoidable sets of the bicyclic inverse semigroup are classified.