Extensions and Congruences of Partial Lattices

Author:

Chajda Ivan1,Länger Helmut23

Affiliation:

1. * Department of Algebra and Geometry , Palacký University Olomouc , 17. listopadu 12 , Olomouc , CZECH REPUBLIC

2. ** Institute of Discrete Mathematics and Geometry , TU Wien , Wiedner Hauptstraße 8-10 , Vienna , AUSTRIA

3. Department of Algebra and Geometry , Palacký University Olomouc , 17. listopadu 12 , Olomouc , CZECH REPUBLIC

Abstract

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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

Reference5 articles.

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