Extensions and Congruences of Partial Lattices
Ivan Chajda, Helmut Länger · Mathematica Slovaca · 2023
Abstract For a partial latticeLthe so-called two-point extension is defined in order to extendLto 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 quotientL/Eof a partial latticeLby a congruenceEonLis again a partial lattice and that the two-point extension ofL/Eis isomorphic to the quotient lattice of the two-point extensionL*ofLby the congruence onL*generated byE. Several illustrative examples are enclosed.