Congruence Lattice

The set of all congruence relations on an algebra, ordered by inclusion, equipped with the lattice operations of meet (intersection) and join (the smallest congruence containing the union, equivalently the congruence generated by the union); this structure is called the congruence lattice and reflects the quotient and equivalence structure compatible with the algebraic operations.

The partially ordered set of all congruence relations on a given algebra, ordered by inclusion; this poset is a lattice under intersection and join (generated equivalence), encoding how quotients relate and combine.