3. Subdirectly irreducible algebras

A subdirect product is said to be trivial if one of the coordinate projections is one-to-one, so that it is an isomorphism from ${\cal A} _ 0$ onto one of the factors.

Similarly, a subdirect representation of ${\cal A}$ is said to be trivial if the image is a trivial subdirect product of the factors. In that case, the factor is isomorphic to ${\cal A}$.

3.1 Definition. An algebra ${\cal A}$ is subdirectly irreducible (SI) if $\vert A\vert>1$ and all subdirect representations of ${\cal A}$ are trivial.

3.2 Theorem (Subdirect Representation Theorem) Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.

For example, every distributive lattice is a subdirect product of two-element chains. (See Application below.)