In algebra, a subdirect irreducible is an algebra that cannot be factored as a subdirect product of "simpler" algebras. Subdirect irreducibles play a somewhat analogous role in algebra to primes in number theory.

Definition

In algebra, a subdirect irreducible (SI), or subdirectly irreducible algebra, is an algebra every subdirect representation of which includes itself (up to isomorphism) as a factor.

Examples

The two-element chain, as either a Boolean algebra, a lattice, or a semilattice, is an SI.
The three-element chain, as either a lattice or a semilattice, is subdirectly representable as a subalgebra of the square of the two-element chain and is therefore not an SI. This representation extends straightforwardly to chains of any cardinality and any order type.
Any finite chain with two or more elements, as a Heyting algebra, is an SI. (The subdirect representation of the three-element chain that worked for lattices and semilattices does not work here because neither 3-chain in the square is closed under complement defined as x → 0.)
The chain of natural numbers together with infinity, as a Heyting algebra, is not an SI, being subdirectly representable as a subalgebra of the direct product of the finite linearly ordered Heyting algebras.
Any cyclic group of prime order is an SI.
The group of integers under addition is not an SI, being subdirectly representable by the cyclic groups of prime order.

Properties

The subdirect representation theorem of universal algebra states that every algebra is subdirectly representable by its subdirectly irreducible quotients. An equivalent definition of "subdirect irreducible" therefore is any algebra A that is not subdirectly representable by those of its quotients not isomorphic to A. (This is not quite the same thing as "its proper quotients" because a proper quotient of A may be isomorphic to A, for example the quotient of the semilattice (Z, min) obtained by identifying just the two elements 3 and 4.)

An immediate corollary is that any variety, as a class closed under homomorphisms, subalgebras, and direct products, is determined by its subdirectly irreducible members, or SIs, since every algebra A in the variety can be constructed as a subalgebra of a suitable direct product of the subdirectly irreducible quotients of A, all of which belong to the variety because A does. For this reason one often studies not the variety itself but just its SIs.

An algebra is subdirectly irreducible if and only if it contains two elements that are identified by every proper quotient. That is, any SI must contain a specific pair of elements witnessing its irreducibility in this way. Given such a witness (a,b) to subdirect irreducibility we say that the SI is (a,b)-irreducible.

Applications

A necessary and sufficient condition for a Heyting algebra to be an SI is for 1 to cover exactly one element (the "penultimate" element). The SI witness is the top pair, and identifying any other pair a, b of elements identifies both a→b and b→a with 1 thereby collapsing everything above those two implications to 1. Hence every finite chain of two or more elements as a Heyting algebra is an SI.

Given any class C of similar algebras, Jonsson's Lemma states that the SIs of the variety HSP(C) generated by C lie in HS(C_SI) where C_SI denotes the class of subdirectly irreducible quotients of the members of C. That is, whereas one must close C under all three of homomorphisms, subalgebras, and direct products to obtain the whole variety, it suffices to close the SIs of C under just homomorphisms and subalgebras to obtain the SIs of the variety. It follows that the SIs of the variety generated by a class of SIs are no larger than the generating SIs, since the quotients and subalgebras of an algebra A are never larger than A itself.