Semidegenerate Congruence-modular Algebras Admitting a Reticulation

Abstract

The reticulation L(R) of a commutative ring R was introduced by Joyal in 1975, then the theory was developed by Simmons in a remarkable paper published in 1980. L(R) is a bounded distributive algebra whose main property is that the Zariski prime spectrum Spec(R) of R and the Stone prime spectrum SpecId (L(R)) of L(R) are homeomorphic. The construction of the lattice L(R) was generalized by Belluce for each unital ring R and the reticulation was defined by axioms. In a recent paper we generalized the Belluce construction for algebras in a semidegenerate congruence-modular variety V. For any algebra A ∈ V we defined a bounded distributive lattice L(A), but in general the prime spectrum Spec(A) of A is not homeomorphic with the prime spectrum SpecId (L(A)). We introduced the quasi-commutative algebras in the variety V (as a generalization of Belluce’s quasi-commutative rings) and proved that for any algebra A ∈ V, the spectra Spec(A) and SpecId (L(A)) are homeomorphic. In this paper we define the reticulation A ∈ V by four axioms and prove that any two reticulations of A are isomorphic lattices. By using the uniqueness of reticulation and other results from the mentioned paper, we obtain a characterization theorem for the algebras A ∈ V that admit a reticulation: A is quasi-commutative if and only if A admits a reticulation. This result is a universal algebra generalization of the following Belluce theorem: a ring R is quasi-commutative if and only if R admits a reticulation. Another subject treated in this paper is the spectral closure of the prime spectrum Spec(A) of an algebra A ∈ V, a notion that generalizes the Belluce spectral closure of the prime spectrum of a ring.