Extensions and congruences of partial lattices

TL;DR

This paper introduces two-point extensions for partial lattices, explores their properties, and establishes a relationship between congruences on partial lattices and their extensions, enabling quotient constructions.

Contribution

It defines two-point extensions for partial lattices, introduces congruences in this context, and proves their compatibility with quotient structures and extensions.

Findings

Two-point extensions extend partial lattices to full lattices.

Congruences on partial lattices relate to homomorphisms and extensions.

Quotients of partial lattices by congruences are preserved under two-point extensions.

Abstract

For a partial lattice L the so-called two-point extension is defined in order to extend L to a lattice. We are motivated by the fact that the one-point extension broadly used for partial algebras does not work in this case, i.e. the one-point extension of a partial lattice need not be a lattice. We describe these two-point extensions and prove several properties of them. We introduce the concept of a congruence on a partial lattice and show its relationship to the notion of a homomorphism and its connections with congruences on the corresponding two-point extension. In particular we prove that the quotient L/E of a partial lattice L by a congruence E on L is again a partial lattice and that the two-point extension of L/E is isomorphic to the quotient lattice of the two-point extension L* of L by the congruence on L* generated by E. Several illustrative examples are enclosed.