Abstract. Let $L_1$ be a finite lattice with an ideal $L_2$. Then the restriction map is a $\{0,1\} $-homomorphism from $\mathop{\rm Con} L_1$ into $\mathop{\rm Con} L_2$. In 1986, the present authors published the converse. If $D_1$ and $D_2$ are finite distributive lattices, and $\varphi \colon D_1 \to D_2$ is a $\{0,1\} $-homomorphism, then there are finite lattices $L_1$ and $L_2$ with an embedding $\eta$ of $L_2$ as an ideal of $L_1$, and there are isomorphisms $\varepsilon_1\colon\mathop{\rm Con} L_1 \to D_1$ and $\varepsilon_2 \colon\mathop{\rm Con} L_2 \to D_2$ such that $\varphi$ is represented as the restriction map of congruences from $L_1$ to $L_2$, up to the two isomorphisms. Let us call a lattice isoform, if for any congruence, all congruence classes are isomorphic lattices. In 2003, G. Grätzer and E. T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of an isoform lattice. In this paper we combine the two results, reproving the 1986 result with isoform lattices.

AMS Subject Classification (1991): 06B10, 06B15

Keyword(s): congruence lattice, congruence-preserving extension, isoform

Received November 12, 2008, and in final form April 12, 2009. (Registered under 6420/2009.)