ACTA issues

Representations of join-homomorphisms of distributive lattices with doubly $2$-distributive lattices

G. Gr├Ątzer, E. T. Schmidt

Acta Sci. Math. (Szeged) 64:3-4(1998), 373-387

Abstract. In the early eighties, A. Huhn proved that if $D$, $E$ are finite distributive lattices and $\psi\colon D \to E$ is a $\{0\} $-preserving join-embedding, then there are finite lattices $K$, $L$ and there is a lattice homomorphism $\varphi\colon K \to L$ such that $\mathop{\rm Con}K$ (the congruence lattice of $K$) is isomorphic to $D$, $\mathop{\rm Con}L$ (the congruence lattice of $L$) is isomorphic to $E$, and the natural induced mapping $\mathop{\rm ext}\varphi\colon \mathop{\rm Con}K \to\mathop{\rm Con}L$ represents $\psi $. The present authors with H. Lakser generalized this result to an arbitrary $\{0\} $-preserving join-homomorphism $\psi $. It was also A. Huhn who introduced the {\it $2$-distributive identity}: $$ x \wedge(y_1 \vee y_2 \vee y_3)= (x \wedge(y_1 \vee y_2)) \vee(x \wedge(y_1 \vee y_3)) \vee(x \wedge(y_2 \vee y_3)). $$ We shall call a lattice {\it doubly $2$-distributive}, if it satisfies the $2$-distributive identity and its dual. In this note, we prove that {\it the lattices $K$ and $L$ in the above result can be constructed as doubly $2$-distributive lattices}.

AMS Subject Classification (1991): 06B10, 06D05

Keyword(s): Lattice, finite, congruence, distributive, join-homomorphism, 2, -distributive

Received May 7, 1998 and in revised form August 25, 1998. (Registered under 3294/2009.)