Mitsch's order and inclusion for binary relations and partitions

TL;DR

This paper characterizes Mitsch's natural partial order on binary relations using relation algebra equations and explores its relationship with inclusion order through lattice structures, also extending to partition monoids.

Contribution

It provides a new algebraic characterization of Mitsch's order and analyzes its relationship with inclusion order within lattice frameworks, including for partition monoids.

Findings

Mitsch's order characterized by algebraic equations

Relationship between Mitsch's order and inclusion order analyzed

Lattice structures for relations and partitions described

Abstract

Mitsch's natural partial order on the semigroup of binary relations is here characterised by equations in the theory of relation algebras. The natural partial order has a complex relationship with the compatible partial order of inclusion, which is explored by means of a sublattice of the lattice of preorders on the semigroup. The corresponding sublattice for the partition monoid is also described.