Noncommutative orders. A preliminary study

TL;DR

This paper explores a categorical approach to linearising partial orders and equivalence relations by replacing sets with coalgebras and Cartesian products with tensor products, providing new definitions and examples.

Contribution

It introduces a novel categorical framework for linearising orders and relations on coalgebras, extending classical definitions to braided monoidal categories.

Findings

Defined orders and relations on coalgebras with explicit examples

Illustrated linearisation using coalgebras spanned by grouplike elements

Proposed definitions applicable beyond vector spaces to other categories

Abstract

The first steps towards linearisation of partial orders and equivalence relations are described. The definitions of partial orders and equivalence relations (on sets) are formulated in a way that is standard in category theory and that makes the linearisation (almost) automatic. The linearisation is then achieved by replacing sets by coalgebras and the Cartesian product by the tensor product of vector spaces. As a result, definitions of orders and equivalence relations on coalgebras are proposed. These are illustrated by explicit examples that include relations on colagebras spanned by grouplike elements (or linearised sets), the diagonal relation, and an order on a three-dimensional non-cocommutative coalgebra. Although relations on coalgebras are defined for vector spaces, all the definitions are formulated in a way that is immediately applicable to other braided monoidal categories.