1. Subdirect products

1.1 Definition. A subdirect product of ${\cal B}$ and ${\cal C}$ is a subalgebra ${\cal A} _ 0$ of ${\cal B} \times {\cal C}$ such that the two coordinate projection maps carry ${\cal A} _ 0$ onto ${\cal B}$ and ${\cal C}$ respectively. In other words, every element of $B$ is used as a coordinate in $A _ 0$ and so is every element of $C$. A heuristic picture is given in Figure .

More generally, the same definition applies for a subalgebra of a direct product over any index set: ${\cal A} \subseteq \Pi _ {\gamma \in \Gamma} {\cal B} _ \gamma$, projection onto each factor.

You can see one virtue of subdirect products: ${\cal A}$ is obtained from ${\cal B}$ and ${\cal C}$, but also you can get from ${\cal A}$ back to ${\cal B}$ and ${\cal C}$ by taking homomorphic images.

Often we say that ${\cal A}$ ``is'' a subdirect product of some other algebras when we really mean that ${\cal A}$ is isomorphic to such a subdirect product.