Varieties, Free Algebras and Equational Logic
Subdirect Products and Subdirect Embeddings
Subalgebras of a direct product that project onto every factor. The construction is weaker than a direct product but available everywhere, and it is the decomposition the subject actually uses.
Learning objectives
- Define subdirect product and subdirect embedding
- Characterise subdirect representations by congruences meeting to Δ
- Explain why subdirect products replace direct products in the general theory
The definition
An algebra A is a subdirect product of a family (Ai)i∈I if A is a subalgebra of ∏i Ai and each projection πi restricted to A is surjective onto Ai.
An embedding α: A → ∏i Ai such that πi ∘ α is surjective for every i.
Surjectivity of the projections is the whole content of the word “subdirect”. Without it, every algebra would be a subalgebra of a product of copies of itself and the notion would be vacuous.
The congruence criterion
The map sends a to the tuple (a/θi)i. It is a homomorphism automatically, surjective onto each factor automatically, and injective precisely when the congruences meet to Δ.
A direct decomposition needs congruences that meet to Δ, join to ∇ and permute. A subdirect decomposition needs only the meet condition. Dropping two of three requirements is what makes subdirect products available everywhere.
Trivial and non-trivial representations
Every algebra has trivial subdirect representations — take the single congruence Δ, giving A as a subdirect product of one copy of itself. The interesting question is when a non-trivial representation exists.
A subdirect representation is trivial if some projection is already an isomorphism — that is, if some θi = Δ. An algebra admitting only trivial representations is called subdirectly irreducible, and those algebras are the subject of the next page.
Examples
| Algebra | Subdirect representation |
|---|---|
| C6, cyclic of order 6 | Subdirect (indeed direct) product of C2 and C3 |
| Any Boolean algebra | Subdirect product of copies of the two-element algebra 2 |
| Any distributive lattice | Subdirect product of copies of the two-element chain |
| The integers as a ring | Subdirect product of the rings Z/pZ is not faithful; Z is subdirectly irreducible as a ring? — in fact Z embeds subdirectly in ∏p Z/pn |
| A vector space of dimension n | Subdirect product of n copies of the field |
That every Boolean algebra is a subdirect product of copies of 2 is the algebraic content of the Stone representation theorem. It is the clearest instance of subdirect decomposition doing real work, and Chapter IV develops it at length.
Frequently asked questions
Is a subdirect product ever equal to the full direct product?
Yes, whenever the subalgebra is all of the product. Direct products are the extreme case of subdirect products, and the notion is a genuine generalisation.
Why require surjectivity onto each factor?
Without it the factors carry no information — one could pad the family with arbitrary algebras that the embedding never reaches. Surjectivity ensures every factor is genuinely a homomorphic image of A.
Source. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, The Millennium Edition — a corrected re-typesetting of Springer-Verlag Graduate Texts in Mathematics 78 (1981). Section II.8, book pages 62-64.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates and reorganises mathematical results; it is not a reproduction of the source text.