2022/01/16 by Ivan Chajda, Chajda, Ivan, Helmut Länger +1
Computer Science · Mathematics · #06A06 #06B05 #06B10 #06B75 #08A55 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rings, Modules, and Algebras #Rough Sets and Fuzzy Logic #math.RA #msc:06A06 #msc:06B05 #msc:06B10 #msc:06B75 #msc:08A55
paper · pdf · doi:10.48550/arxiv.2201.06046
arxiv created 2022/01/16 · openalex publication_date 2022/01/16 · arxiv updated 2022/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.