Transferring Davey`s Theorem on Annihilators in Bounded Distributive Lattices to Modular Congruence Lattices and Rings

TL;DR

This paper extends Davey's Theorem on annihilators from bounded distributive lattices to congruence lattices of semiprime algebras in certain varieties, showing the preservation of key properties across structures.

Contribution

It transfers Davey's Theorem to congruence lattices of semiprime algebras and rings, demonstrating the invariance of conditions under various algebraic constructions.

Findings

Congruence lattices of semiprime algebras satisfy Davey's conditions.

These conditions are preserved under finite direct products.

Similar equivalences hold for elements in rings and residuated lattices.

Abstract

Congruence lattices of semiprime algebras from semi--degenerate congruence--modular varieties fulfill the equivalences from B. A. Davey`s well--known characterization theorem for $m$--Stone bounded distributive lattices, moreover, changing the cardinalities in those equivalent conditions does not change their validity. I prove this by transferring Davey`s Theorem from bounded distributive lattices to such congruence lattices through a certain lattice morphism and using the fact that the codomain of that morphism is a frame. Furthermore, these equivalent conditions are preserved by finite direct products of such algebras, and similar equivalences are fulfilled by the elements of semiprime commutative unitary rings and, dualized, by the elements of complete residuated lattices.