Subdirectly irreducible algebras

Abstract

Abstract An algebra L is said to be subdirectly irreducible if it has a smallest nontrivial congruence; i.e. a congruence α such that , ϑ ≥ a for all ϑ ∈ Con L with , ϑ ≠ w. Such a congruence a is called the monolith of Con L. The importance of such algebras is shown in a classic theorem of Birkhoff [2] which states that in an equational class of algebras every algebra can be embedded in a direct product of subdirectly irreducible algebras. An immediate consequence of the above definition is that if L is subdirectly irreducible then in Con L the trivial congruence w is ∧-irreducible. A particularly important case of a subdirectly irreducible algebra is a simple algebra, namely one for which the lattice of congruences is the two-element chain {w, i}.