Tue, 10 Feb 2026, 1:25 pm MT
Commutator lattices [7] are complete lattices endowed with an additional binary operation, called commutator,
which is commutative, smaller than its arguments and completely distributive w.r.t. the join. They serve as abstrac-
tions for congruence lattices of algebras whose term–condition commutators are commutative and distributive w.r.t.
arbitrary joins, in particular for congruence lattices of members of congruence–modular varieties endowed with the
modular commutator. The (minimal) prime spectrum of a commutator lattice is its set of (minimal) prime elements
w.r.t. the commutator, and its topological structure turns out to be particularly useful.
Using the Stone topology on the prime spectrum of a congruence lattice, we have constructed and studied the
reticulation of a universal algebra in [2, 3]. In [4] we have studied the Stone, as well as the flat topology on the
minimal prime spectrum of congruences, then used these topologies to investigate congruences in extensions of universal
algebras, generalizing results from [1] on ideals in ring extensions.
In [5, 6] we have developed this theory for commutator lattices, using complete join–semilattice morphisms between
commutator lattices as an abstraction for the action on congruences of morphisms between similar algebras. When
these morphisms are embeddings, we obtain the particular case in [4] of congruence extensions. My talk will focus on
the abstract case in [5, 6].
Joint work with George Georgescu and Leonard Kwuida.
References
[1] P. Bhattacharjee, K.M. Dress, W.W. McGovern, Extensions of Commutative Rings, Topology and Its Applications 158
(2011), 1802–1814.
[2] G. Georgescu, C. Muresan, The Reticulation of a Universal Algebra, Scientific Annals of Computer Science 28 (1) (2018),
67–113.
[3] G. Georgescu, L. Kwuida, C. Muresan, Functorial Properties of the Reticulation of a Universal Algebra, Journal of Applied
Logics. Special Issue: Multiple–Valued Logics and Applications 8 (5) (2021), 1123–1168.
[4] G. Georgescu, L. Kwuida, C. Muresan, Congruence Extensions in Congruence–modular Varieties, Axioms 13 (12) (2024),
824.
[5] G. Georgescu, L. Kwuida, C. Muresan, Stone and Flat Topologies on the Minimal Prime Spectrum of a Commutator
Lattice, Axioms 14 (11) (2025), 803.
[6] G. Georgescu, L. Kwuida, C. Muresan, An Abstraction of Congruence Extensions through Complete Join–semilattice
Morphisms between Commutator Lattices, submitted.
[7] C. Muresan, Stone Commutator Lattices and Baer Rings, Discussiones Mathematicae – General Algebra and Applications
42 (1) (2022), 51–96.