Subdirectly irreducible algebra

Summary

A subdirectly irreducible algebra is a nontrivial algebra that cannot be split as a subdirect product in a nontrivial way. In every subdirect representation of it, one of the projections is already an isomorphism. Equivalently, among its congruences other than equality there is a smallest one, called the monolith. By Birkhoff's subdirect representation theorem, these algebras are the building blocks of all algebras, and a variety is determined by its subdirectly irreducible members.