Subdirect product

In universal algebra, the problem arises that all ( universal ) algebras can not be represented as a direct product of irreducible algebras. The problem is solved in the so-called subdirect product, a certain kind of a subalgebra of a direct product. The first representation theorem of Garrett Birkhoff then states that every algebra can be written subdirectly irreducible algebras as subdirect product.

Definition

There are algebras of the same type, that is of the same algebraic structure, and an index family. A subalgebra is called a subdirect product of, if for all, with the canonical projection called.

Subdirect irreducibility

An embedding is called subdirect representation of if subdirect product of.

Is called subdirectly irreducible if such exist, for any subdirect representation that is an isomorphism.

Motivation

The fact that an algebra normally can not be represented as a direct product of irreducible algebras, the following example shows: A Boolean algebra is exactly then directly or subdirectly irreducible if and only if. A countably infinite Boolean algebra is given by with support amount. This can not possibly be a direct product of two-element algebras, since such a product is either finite or uncountable.

Representation theorem of Birkhoff

Each algebra is isomorphic to a subdirect product subdirectly irreducible algebras of the same type. The representation as a subdirect product is not unique.

Example

Above mentioned Boolean algebra has for example the following subdirect representation:

With