The Reticulation of a Universal Algebra

Abstract

The reticulation of an algebra $A$ is a bounded distributive lattice $\mathcal{L}(A)$ whose prime spectrum of filters or ideals is homeomorphic to the prime spectrum of congruences of $A$, endowed with the Stone topologies. We have obtained a construction for the reticulation of any algebra $A$ from a semi-degenerate congruence-modular variety $\mathcal{C}$ in the case when the commutator of $A$, applied to compact congruences of $A$, produces compact congruences, in particular when $\mathcal{C}$ has principal commutators; furthermore, it turns out that weaker conditions than the fact that $A$ belongs to a congruence-modular variety are sufficient for $A$ to have a reticulation. This construction generalizes the reticulation of a commutative unitary ring, as well as that of a residuated lattice, which in turn generalizes the reticulation of a BL-algebra and that of an MV-algebra. The purpose of constructing the reticulation for the algebras from $\mathcal{C}$ is that of transferring algebraic and topological properties between the variety of bounded distributive lattices and $\mathcal{C}$, and a reticulation functor is particularily useful for this transfer. We have defined and studied a reticulation functor for our construction of the reticulation in this context of universal algebra.

29 pages

References in corpus (2)

• Taking Prime, Maximal and Two--class Congruences Through Morphisms
• The Commutator as Least Fixed Point of a Closure Operator