Stone and Flat Topologies on the Minimal Prime Spectrum of a Commutator Lattice

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not even need to be enforced on their congruence lattices, let alone on those of the members of the variety they generate. Commutator lattices, defined by Czelakowski in 2008, are commutative multiplicative lattices having as prototype the algebraic structure of the congruence lattice of a such an algebra. Considering the prime elements with respect to the commutator operation, we obtain algebraic characterizations for minimal primes, then study the Stone and flat topologies on the set of minimal primes in a commutator lattice. We also prove abstract versions of congruence extension properties, actually of the general case of arbitrary morphisms instead of algebra embeddings, by means of complete join–semilattice morphisms between commutator lattices. We thus obtain abstractions for our results on congruence lattices and generalizations for results on frames and quantales, but also further cases in which these results hold. Furthermore, we investigate the lattice structures of these topologies as sublattices of the power sets of the sets of (minimal) primes.