Subdirectly Irreducible Algebra

An algebra is subdirectly irreducible (SI) if it cannot be represented nontrivially as a subdirect product of other algebras; equivalently, it has a least nontrivial congruence (the monolith), i.e., among congruences distinct from the diagonal there is a unique minimal one.

An algebra that cannot be represented as a nontrivial subdirect product of other algebras; equivalently (in varieties) it has a unique minimal nontrivial congruence (the monolith).