Quotient Algebra - Congruence Lattice

Congruence Lattice

For every algebra on the set A, the identity relation on A, and are trivial congruences. An algebra with no other congruences is called simple.

Let be the set of congruences on the algebra . Because congruences are closed under intersection, we can define a meet operation: by simply taking the intersection of the congruences .

On the other hand, congruences are not closed under union. However, we can define the closure of any binary relation E, with respect to a fixed algebra, such that it is a congruence, in the following way: . Note that the (congruence) closure of a binary relation depends on the operations in, not just on the carrier set. Now define as .

For every algebra, with the two operations defined above forms a lattice, called the congruence lattice of .