New Examples of Dimension Zero Categories

TL;DR

This paper demonstrates that certain categories derived from finite idempotent semirings, including the category of finite sets with relations, are dimension zero over any infinite field, meaning all finitely generated representations have finite length.

Contribution

It introduces new examples of dimension zero categories, particularly those arising from finite idempotent semirings, expanding understanding of their representation theory.

Findings

Categories from finite idempotent semirings are dimension zero over infinite fields

The category of finite sets with relations is dimension zero over any infinite field

Finitely generated representations in these categories have finite length

Abstract

We say that a category $\mathscr{D}$ is dimension zero over a field $F$ provided that every finitely generated representation of $\mathscr{D}$ over $F$ is finite length. We show that $\textrm{Rel}(R)$, a category that arises naturally from a finite idempotent semiring $R$, is dimension zero over any infinite field. One special case of this result is that $\textrm{Rel}$, the category of finite sets with relations, is dimension zero over any infinite field.