Varieties, Free Algebras and Equational Logic

Class Operators H, S, P and their Composition

The operators taking homomorphic images, subalgebras and products of a class, the inclusions among their composites, and the identity HSP that computes the generated variety.

Category Engineering / MathematicsSource II.9Pages 66-67Reading 2 minReviewed 2026-08-07

Learning objectives

Define H, S, P and I and state their basic properties
Prove the key inclusions SH ≤ HS, PS ≤ SP, PH ≤ HP
Show that HSP is a closure operator

The operators

I(K): all algebras isomorphic to a member of K
H(K): all homomorphic images of members of K
S(K): all subalgebras of members of K
P(K): all direct products of families from K
PS(K): all subdirect products of families from K
PU(K): all ultraproducts of families from K

Each operator is extensive (K ≤ O(K)) and monotone. H, S and I are idempotent; P is idempotent once I is absorbed, since a product of products is isomorphic to a single product.

Convention on empty productsP(K) is taken to include the empty product, the one-element algebra. This makes P extensive on the empty class and keeps the trivial algebra in every variety.

The commuting inclusions

Key inclusions

For any class K: SH(K) ≤ HS(K), PS(K) ≤ SP(K), PH(K) ≤ HP(K).

Each says an operator can be pushed past another in one direction. Taking SH ≤ HS as an example: a subalgebra B of a homomorphic image α(A) is the image under α of the preimage α⁻¹(B), which is a subalgebra of A. So B ∈ HS(K).

The inclusions are strict in general

None of the three reverses. A homomorphic image of a subalgebra need not be a subalgebra of a homomorphic image, and so on. Getting the direction right is the whole content of the HSP computation.

HSP is a closure operator

HSP computes the generated varietyHSP is a closure operator on classes of algebras of a fixed type, and HSP(K) is the smallest class containing K that is closed under H, S and P.

Idempotence is the substantial claim. Using the inclusions:

HSPHSP ≤ HSHPSP ≤ HHSPSP ≤ HSSPP ≤ HSP

Each step applies one of the three inclusions or an idempotence, pushing H leftward and P rightward until the composite collapses. That this works is the reason HSP, and not some longer word in the operators, is the right expression.

Consequence

There is no need to iterate. Applying H, S and P once each in that order already produces a class closed under all three. This is what makes Birkhoff's theorem computable in principle.

Subdirect products in terms of the operators

The subdirect product operator satisfies P_S ≤ SP, since a subdirect product is by definition a subalgebra of a product. Birkhoff's subdirect representation theorem can then be written as a statement about operators:

Every algebra lies in P_S(K_SI), where K_SI is the class of subdirectly irreducible members

This operator formulation is the one used in Chapter IV when Jónsson's lemma bounds the subdirectly irreducibles by HSP_U(K).

Frequently asked questions

Why is HSP the right order rather than PSH or SHP?

Because the inclusions push H to the left and P to the right. Any word in the operators reduces to HSP, so HSP is the canonical form — and applying them in a different order may fail to give a closed class in one pass.

Is HSP(K) always a proper class?

Yes, for non-trivial K, since it contains algebras on arbitrarily large sets. This is why the subject needs classes rather than sets.

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.9, book pages 66-67.

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