A categorification of biclosed sets of strings

TL;DR

This paper explores the structure of biclosed sets of strings in gentle algebras, establishing their lattice properties and connections to torsion and wide shadows, thus advancing the categorification of these algebraic structures.

Contribution

It introduces the lattice of torsion shadows and wide shadows, proving their isomorphism with biclosed sets and shard intersection orders, respectively, generalizing previous results.

Findings

Biclosed sets of strings form a congruence-uniform lattice.

The lattice of biclosed sets is isomorphic to a lattice of torsion shadows.

The shard intersection order of biclosed sets is isomorphic to a lattice of wide shadows.

Abstract

We consider the closure space on the set of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows. Finite congruence-uniform lattices admit an alternate partial order known as the shard intersection order. In many cases, the shard intersection order of a congruence-uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra. Analogous to torsion shadows, we introduce wide shadows, and prove that the shard intersection order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows.