ACTA issues

## Minimal representations of a finite distributive lattice by principal congruences of a lattice

George Grätzer, Harry Lakser

Acta Sci. Math. (Szeged) 85:1-2(2019), 69-96
60/2017

 Abstract. Let the finite distributive lattice $D$ be isomorphic to the congruence lattice of a finite lattice $L$. Let $Q$ denote those elements of $D$ that correspond to principal congruences under this isomorphism. Then $Q$ contains $0,1 \in D$ and all the join-irreducible elements of $D$. If $Q$ contains exactly these elements, we say that $L$ is a minimal representation of $D$ by principal congruences of the lattice $L$. We characterize finite distributive lattices $D$ with a minimal representation by principal congruences with the property that $D$ has at most two dual atoms. DOI: 10.14232/actasm-017-060-9 AMS Subject Classification (1991): 06B10, 06A06 Keyword(s): finite lattice, principal congruence, ordered set Received October 10, 2017, and in revised form November 26, 2017. (Registered under 60/2017.)