Simultaneous Congruence Representations: a Special Case

We study the problem of representing a pair of algebraic lattices, L1 and L0, as Con(A1) and Con(A0), respectively, with A1 an algebra and A0 a subalgebra of A1, and we provide such a representation in a special case. Back in 1971, Ervin Fried posed the problem of representing a pair of algebraic lattices, L1 and L0, as Con(A1) and Con(A0), respectively, with A1 an algebra and A0 a subalgebra of A1. This seems to be a very hard problem in general. Note that if Con(A1) has only one element, then Con(A0) has only one element. It is known that if L is an algebraic lattice having a compact 1, then L is isomorphic to the congruence lattice of a groupoid A = 〈A, ·〉, where · is a binary operation. We settle Fried’s question positively in the case that both L1 and L0 have compact 1’s. Theorem 1. Suppose L1 and L0 are algebraic lattices and L1 has at least 2 elements and L1 and L0 both have compact 1’s. Then there is a groupoid A1 = 〈A1, ·〉 and a subgroupoid A0, so that Li is isomorphic to Con(Ai), for i = 0, 1. This solves the simultaneous representation problem for pairs of finite lattices, as we point out in the Corollary 2. Suppose L1 and L0 are finite lattices and L1 has at least 2 elements. Then there is a groupoid A1 = 〈A1, ·〉 and a subgroupoid A0, so that Li is isomorphic to Con(Ai), for i = 0, 1. Recall that a semilattice homomorphism is 0–separating iff it is 0–preserving and sends nonzero elements to nonzero elements. We suppose A1 is an algebra and A0 is a subalgebra of A1. Let S be a set of pairs. To simplify notation we will let [S]Ai denote the congruence relation of Ai generated by S. Let Θ be a congruence relation of A0. Then the map which sends Θ −→ [Θ]A1 is a complete, 0–separating join homomorphism from Con(A0) into Con(A1) sending compact elements to compact elements. Such a homomorphism is determined by its action on the semilattice of compact elements. Since the only subdirectly irreducible semilattice is the two element one, there is always at least one 0–separating homomorphism from any given semilattice to any other as long as the latter has at least two elements. More precisely, the theorem we will prove is the following. Theorem 3. Suppose L1 and L0 are algebraic lattices and L1 has at least 2 elements and L1 and L0 both have compact 1’s. Suppose also that γ is a complete, 0–separating, 1–preserving join homomorphism from L0 to L1 sending compact elements to compact elements. Then there exists a groupoid A1 = 〈A1, ·〉 and a subgroupoid A0 and an isomorphism σi from Li onto Con(Ai) such that [σ0(x)]A1 = σ1(γ(x)) for any x ∈ L0. Date: January 12, 2005. 1991 Mathematics Subject Classification: 06B15, 08A30.