Explore: Congruence Lattices

congruence lattice is true for all distributive lattices with at most ℵ1 compact elements, but F. Wehrung provided a counterexample for distributive lattices with

complemented lattices, Heyting algebras etc. Furthermore, every congruence-permutable algebra is congruence-modular, i.e. its lattice of congruences is modular

the same is true for finite lattices and finite algebras. That is, does every finite lattice occur as the congruence lattice of a finite algebra? In 1980

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple

semigroup congruence ~ induces congruence classes [a]~ = {x ∈ S | x ~ a} and the semigroup operation induces a binary operation ∘ on the congruence classes:

of congruence relations, the quotient lattices need not be distributive or modular again. In other words, the varieties of distributive lattices and

171–199, MR 0005721 Kearnes, Keith; Kiss, Emil W. (2013). The Shape of Congruence Lattices. American Mathematical Soc. p. 3. ISBN 978-0-8218-8323-5. Ore, Øystein

rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object

of groups, consist of congruence-permutable algebras, but some, like the variety of lattices, have members that are not congruence-permutable. Given an

notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively