ACTA issues

Characterizing representability by principal congruences for finite distributive lattices with a join-irreducible unit element

George Gr├Ątzer

Acta Sci. Math. (Szeged) 83:3-4(2017), 415-431
36/2017

Abstract. For a finite distributive lattice $D$, let us call $Q \ci D$ \emph{principal congruence representable}, if there is a finite lattice $L$ such that the congruence lattice of $L$ is isomorphic to $D$ and the principal congruences of $L$ correspond to $Q$ under this isomorphism. We find a necessary condition for representability by principal congruences and prove that for finite distributive lattices with a join-irreducible unit element this condition is also sufficient.



DOI: 10.14232/actasm-017-036-7

AMS Subject Classification (1991): 06B10

Keyword(s): congruence lattice, principal congruence, join-irreducible congruence, finite distributive lattice, principal congruence representable set


Received May 22, 2017, and in revised form August 1, 2017. (Registered under 36/2017.)