Functorial Properties of the Reticulation of a Universal Algebra

TL;DR

This paper explores the properties of the reticulation functor in universal algebra, characterizing when it preserves morphism properties and how algebraic-topological features transfer between algebras and distributive lattices.

Contribution

It characterizes morphisms that admit reticulation images and identifies varieties where reticulation functors preserve morphism injectivity and other properties.

Findings

Reticulation functors are characterized for various algebraic varieties.

Conditions like Going Up, Going Down, and Lying Over transfer through reticulation.

Reticulation preserves injectivity of morphisms under certain varietal properties.

Abstract

The {\em reticulation} of an algebra $A$ is a bounded distributive lattice whose prime spectrum of ideals (or filters), endowed with the Stone topology, is homeomorphic to the prime spectrum of congruences of $A$, with its own Stone topology. The reticulation allows algebraic and topological properties to be transferred between the algebra $A$ and bounded distributive lattices, a transfer which is facilitated if we can define a {\em reticulation functor} from a variety containing $A$ to the variety of (bounded) distributive lattices. In this paper, we continue the study of the reticulation of a universal algebra initiated in \cite{retic}, where we have used the notion of a prime congruence introduced through the term condition commutator. We characterize morphisms which admit an image through the reticulation and investigate the kinds of varieties that admit reticulation functors; we prove that these include semi--degenerate congruence--distributive varieties with the Compact Intersection Property and semi--degenerate congruence--distributive varieties with congruence intersection terms, as well as generalizations of these, and additional varietal properties ensure that the reticulation functors preserve the injectivity of morphisms. We also study the property of morphisms of having an image through the reticulation in relation to another property, involving the complemented elements of congruence lattices, exemplify the transfer of properties through the reticulation with conditions Going Up, Going Down, Lying Over and the Congruence Boolean Lifting Property, and illustrate the applicability of such a transfer by using it to derive results for certain types of varieties from properties of bounded distributive lattices.