# Subdirect product

of algebraic systems

A special type of subsystem in a direct (Cartesian) product of systems (cf. Direct product). Let , , be a family of algebraic systems of the same type and let be the direct product of these systems with the projections , . An algebraic system of the same type is called a subdirect product of the systems if there is an imbedding such that the homomorphisms , , are surjective. Sometimes, by a subdirect product is meant any system that is isomorphic to a subsystem of the direct product; then the systems that satisfy the above condition are called special subdirect products. In the theories of rings and modules, a subdirect product is also called a subdirect sum. A subdirect product (subdirect sum) is denoted by (, respectively).

The following conditions are equivalent: a) the system is a subdirect product of the systems , ; b) there exists a separating family of surjective homomorphisms , ; c) there exists a family of congruences , , of the system such that the intersection of these congruences is the identity congruence and for each . Any universal algebra is a subdirect product of subdirectly irreducible algebras.

From the category-theoretic point of view, the concept of a subdirect product is dual to the concept of the regular product of algebraic systems containing zero (one-element) subsystems.