Subdirect irreducible

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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.

1 Definition
2 Examples
3 Properties
4 Applications

[edit] Definition

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

[edit] Examples

The two-element chain, as either a Boolean algebra, a Heyting algebra, a lattice, or a semilattice, is subdirectly irreducible.
Any finite chain with two or more elements, as a Heyting algebra, is subdirectly irreducible. (This is not the case for chains of three or more elements as either lattices or semilattices, which are subdirectly reducible to the two-element chain. The difference with Heyting algebras is that a → b need not be comparable with a under the lattice order even when b is.)
Any finite cyclic group of order a power of a prime is subdirectly irreducible. (One weakness of the analogy between subdirect irreducibles and prime numbers is that the integers are subdirectly representable by any infinite family of nonisomorphic prime-power cyclic groups, e.g. just those of order a Mersenne prime assuming there are infinitely many.)
A vector space is subdirectly irreducible if and only if it has dimension one.

[edit] 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 "by 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, 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 subdirect irreducibles.

An algebra A is subdirectly irreducible if and only if it contains two elements that are identified by every proper quotient, equivalently, if and only if its lattice Con A of congruences has a least nonidentity element. That is, any subdirect irreducible 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 subdirect irreducible is (a,b)-irreducible.

Given any class C of similar algebras, Jónsson's Lemma states that the subdirect irreducibles 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 subdirect irreducibles of C under just homomorphic images (quotients) and subalgebras to obtain the subdirect irreducibles of the variety.

[edit] Applications

A necessary and sufficient condition for a Heyting algebra to be subdirectly irreducible is for there to be a greatest element strictly below 1. The witnessing pair is that element and 1, 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 subdirectly irreducible.

By Jónsson's Lemma the subdirect irreducibles of the variety generated by a class of subdirect irreducibles are no larger than the generating subdirect irreducibles, since the quotients and subalgebras of an algebra A are never larger than A itself. Hence the subdirect irreducibles in the variety generated by a finite linearly ordered Heyting algebra H must be just the nondegenerate quotients of H, namely all smaller linearly ordered nondegenerate Heyting algebras.