Fiat categorification of the symmetric inverse semigroup IS_n and the semigroup F^*_n

TL;DR

This paper constructs two fiat 2-categories that categorify the semigroup algebras of the symmetric inverse semigroup and its maximal factorizable subsemigroup, extending the framework of categorification in semigroup theory.

Contribution

It introduces new fiat 2-categories as categorifications of the semigroup algebras related to the symmetric inverse semigroup and its dual, expanding the categorification approach.

Findings

Constructed fiat 2-categories for symmetric inverse semigroup

Provided categorifications for semigroup algebras of specific semigroups

Extended the framework of fiat categorification in semigroup theory

Abstract

Starting from the symmetric group $S_n$, we construct two fiat $2$-categories. One of them can be viewed as the fiat "extension" of the natural $2$-category associated with the symmetric inverse semigroup (considered as an ordered semigroup with respect to the natural order). This $2$-category provides a fiat categorification for the integral semigroup algebra of the symmetric inverse semigroup. The other $2$-category can be viewed as the fiat "extension" of the $2$-category associated with the maximal factorizable subsemigroup of the dual symmetric inverse semigroup (again, considered as an ordered semigroup with respect to the natural order). This $2$-category provides a fiat categorification for the integral semigroup algebra of the maximal factorizable subsemigroup of the dual symmetric inverse semigroup.