On the Combinatorial Inverse Monoid IO3

TL;DR

This paper computes the Mobius category and function for the combinatorial inverse monoid IO3, which consists of order-preserving partial bijections on a three-element set, providing insights into its algebraic structure.

Contribution

It introduces the computation of the Mobius category and function for IO3, a specific combinatorial inverse monoid, expanding understanding of its algebraic and combinatorial properties.

Findings

Mobius category of IO3 computed

Mobius function of IO3 determined

Provides structural insights into IO3

Abstract

In this paper we have compute the Mobius category and the Mobius function of the combinatorial inverse monoid IO3 of all order preserving partial bijections on the set M3={1,2,3}. This category is the reduced standard division category CF(IO3) relative to an idempotent transversal F of the D-classes of IO3.