Partitioned binary relations

TL;DR

This paper introduces the category of partitioned binary relations, demonstrating its inclusion of many classical diagram categories, and explores its deformations and factorizations.

Contribution

It defines the category of partitioned binary relations, shows its relation to classical diagram categories, and constructs a one-parameter deformation with a factorization result.

Findings

Contains classical diagram categories like Brauer and Temperley-Lieb

Constructs a one-parameter deformation of the category

Provides a factorization of partitioned binary relations

Abstract

We define the category of partitioned binary relations and show that it contains many classical diagram categories, including categories of binary relations, maps, injective maps, partitions, (oriented) Brauer diagrams and (oriented) Temperley-Lieb diagrams. We construct a one-parameter deformation of the category of partitioned binary relations and show that it gives rise to classical one-parameter deformations of partition, Brauer and Temperley-Lieb categories. Finally, we describe a factorization of partitioned binary relations into a product of certain idempotents and pairs of usual binary relations.