Subdirectly irreducible algebra
Open-knowledge reference entry
In the branch of mathematics known as universal algebra (and in its applications), a subdirectly irreducible algebra is an algebra that cannot be factored as a subdirect product of "simpler" algebras. Subdirectly irreducible algebras play a somewhat analogous role in algebra to primes in number theory.
Definition
A universal algebra A is said to be subdirectly irreducible when A has more than one element, and when any subdirect representation of A includes (as a factor) an algebra isomorphic to A, with the isomorphism being given by the projection map.
Examples
The two-element chain, as either a Boolean algebra, a Heyting algebra, a lattice, or a semilattice, is subdirectly irreducible. In fact, the two-element chain is the only subdirectly irreducible distributive lattice.
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 (i.e. any cyclic p-group) 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.
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