Revisiting the representation theorem of finite distributive lattices with principal congruences

2021/04/28 by George Grätzer, Grätzer, G., H. Lakser +1
Computer Science · #06B10 #Advanced Algebra and Logic #FOS: Mathematics #Rings and Algebras (math.RA) #Rough Sets and Fuzzy Logic #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2104.13835

openalex publication_date 2021/04/28 · openalex created_date 2021/05/10 · openalex updated_date 2026/07/28

Abstract

A classical result of R. P. Dilworth states that every finite distributive lattice D can be represented as the congruence lattice of a finite lattice~L. A~sharper form was published in G.~Grätzer and E. T. Schmidt in 1962, adding the requirement that all congruences in L be principal. Another variant, published in 1998 by the authors and E. T. Schmidt, constructs a planar semimodular lattice L. In this paper, we merge these two results: we construct L as a planar semimodular lattice in which all congruences are principal. This paper relies on the techniques developed by the authors and E. T. Schmidt in the 1998 paper.

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