Semilattices with a congruence

TL;DR

This paper demonstrates that the category of specialization semilattices is equivalent to the category of semilattices with a congruence, providing a new perspective on their structural relationships.

Contribution

It establishes an isomorphism between specialization semilattices and semilattices with a congruence, linking these concepts through categorical equivalence.

Findings

Category of specialization semilattices is isomorphic to semilattices with a congruence

Provides an internal characterization of surjective homomorphisms in relational systems

Shows equivalence to the category of semilattice epimorphisms

Abstract

A specialization semilattice is a semilattice together with a coarser preorder satisfying a compatibility condition. We show that the category of specialization semilattices is isomorphic to the category of semilattices with a congruence, hence equivalent to the category of semilattice epimorphisms. Guided by the above example, we recall an ``internal'' characterization of surjective homomorphisms between general relational systems.