Subdirect product

of algebraic systems

A special type of subsystem in a direct (Cartesian) product of systems (cf. Direct product). Let $ A _ {i} $, $ i \in I $, be a family of algebraic systems of the same type and let $ A = \prod _ {i \in I } A _ {i} $ be the direct product of these systems with the projections $ \rho _ {i} : A \rightarrow A _ {i} $, $ i \in I $. An algebraic system $ B $ of the same type is called a subdirect product of the systems $ A _ {i} $ if there is an imbedding $ m : B \rightarrow A $ such that the homomorphisms $ \rho _ {i} m $, $ i \in I $, 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 $ \prod _ {i \in I } ^ {s } A _ {i} $( $ \sum _ {i \in I } ^ {s } A _ {i} $, respectively).

The following conditions are equivalent: a) the system $ B $ is a subdirect product of the systems $ A _ {i} $, $ i \in I $; b) there exists a separating family of surjective homomorphisms $ f _ {i} : B \rightarrow A _ {i} $, $ i \in I $; c) there exists a family of congruences $ \rho _ {i} $, $ i \in I $, of the system $ B $ such that the intersection of these congruences is the identity congruence and $ B/ \rho _ {i} \simeq A _ {i} $ for each $ i \in I $. 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.

Comments

An algebra $ B $ is called subdirectly irreducible if, in any representation of $ B $ as a subdirect product $ \prod _ {i \in I } ^ {s} A _ {i} $, one of the homomorphisms $ B \rightarrow A _ {i} $ is an isomorphism (equivalently, if the identity congruence on $ B $ is not representable as an intersection of strictly larger congruences). The theorem that every algebra is representable as a subdirect product of subdirectly irreducible algebras is due to G. Birkhoff [a1]; its usefulness stems from the fact that, in many familiar varieties, the subdirectly irreducible algebras are few in number and can easily be described explicitly. For example, the only subdirectly irreducible Boolean algebra is the two-element chain.

References

[a1]	G. Birkhoff, "Subdirect unions in universal algebra" Bull. Amer. Math. Soc. , 50 (1944) pp. 764–768
[a2]	P.M. Cohn, "Universal algebra" , Reidel (1981)