Core Universal Algebra

Class Operators H, S, P and the Definition of a Variety

Six operators, one composite. HSP is a closure operator, and the class it produces is the variety generated — which by Birkhoff's theorem is exactly the equational class.

Engineering · Mathematics4 min readKV-MATH-0218

Learning objectives

Define each class operator and compute simple composites.
State the standard inclusions such as SH ≤ HS and PH ≤ HP.
Show HSP is idempotent and hence a closure operator.
Identify varieties as the classes closed under H, S and P.
Anticipate the HSP theorem's identification of varieties with equational classes.

01The operators

Class operators on classes of algebras of a fixed type
Operator	Meaning
I(K)	all isomorphic copies of members of K
S(K)	all subalgebras of members of K
H(K)	all homomorphic images of members of K
P(K)	all direct products of families of members of K
P_S(K)	all subdirect products of families of members of K
P_U(K)	all ultraproducts of families of members of K

Each operator is extensive and monotone. I is idempotent trivially; so are S, H and P individually. The interest lies entirely in how they compose, because a class closed under all three is a variety and the composite that produces such closure is not obvious a priori.

02Composition relations

The operators do not commute, but they satisfy inclusions that let any composite be reduced to the canonical form HSP.

SH ≤ HS PH ≤ HP PS ≤ SP P_S ≤ SP

Read: a subalgebra of a homomorphic image is a homomorphic image of a subalgebra, and so on. Each inclusion is proved by an explicit construction.

Key resultHSP absorbs everything

Using the inclusions repeatedly, any composite of I, S, H and P is contained in HSP. In particular HSP is idempotent: HSP(HSP(K)) = HSP(K). An extensive, monotone, idempotent operator is a closure operator, so HSP is the closure operator whose closed classes are the varieties.

03Proving SH ≤ HS

ProcedureA subalgebra of a homomorphic image is a homomorphic image of a subalgebra

in: C ∈ SH(K) → out: C ∈ HS(K)

input: surjective homomorphism α : A → B, subalgebra C ≤ B
form the preimage D := α⁻¹(C) ⊆ A
D is a subuniverse: operations applied within D land in α⁻¹ of C, which is D
restrict α to D, giving α|D : D → C
α|D is surjective because α is surjective onto B and C ⊆ B
therefore C ∈ H(S({A})) whenever C ∈ S(H({A}))

The preimage construction is the whole proof. Caveat: the reverse inclusion HS ≤ SH is false — a homomorphic image of a subalgebra need not embed in a homomorphic image of the whole, so the order in HSP is not arbitrary.

The asymmetry is worth internalising. HSP works; SPH does not, and there is no rearrangement that makes it work. The canonical order is forced by the direction of the inclusions.

04Varieties

A variety is a class of algebras of a fixed type closed under H, S and P. Equivalently, a class K with HSP(K) = K.

Closure

Under three operators

H, S and P. Closure under I is implied by closure under S, since isomorphic copies are subalgebras up to identification.

Generation

V(K) = HSP(K)

The variety generated by a class is the smallest variety containing it, and equals HSP(K) by idempotency.

Trivial members

Always present

Any variety contains the one-element algebra, since it is a homomorphic image of any member, and the empty product is the one-element algebra.

Varieties are proper classes, as noted in the preliminaries. They are also closed under the formation of free algebras, which is what makes the equational theory available and is developed in the next stream.

05Towards the HSP theorem

One direction is easy
An equationally defined class is closed under H, S and P, because satisfaction of an identity is preserved by all three constructions. This is a direct verification.
The converse is the theorem
If K is closed under H, S and P then K is defined by the set of equations holding in K. The proof uses free algebras: a free algebra for the equational theory of K must itself lie in K.
The bridge is the free algebra
Constructing F_K(X) as a quotient of the term algebra by the congruence of K-identities, and showing it lies in K, is the crux.
Consequence
Varieties and equational classes coincide exactly. Syntax and closure are the same notion viewed from two sides — a Galois connection, as set up in the closure operator page.

NoteWhere this goes next

The Terms, Free Algebras and Equational Logic stream constructs term algebras and free algebras and completes this proof. The present page establishes only that HSP is the right closure operator; that its closed classes are the equational ones is the theorem to come.

Frequently asked

Is P(K) closed under taking the empty product?

By convention yes, and the empty product is the one-element algebra. This is why every variety contains a trivial algebra, and it is occasionally a nuisance in statements that would prefer to exclude it. Some texts define P to exclude the empty index set; check before comparing results.

Why is the canonical order HSP rather than PSH?

Because the inclusions run SH ≤ HS and PH ≤ HP and PS ≤ SP, all pushing H leftwards and P rightwards. Composing in the order HSP therefore absorbs any other composite. The reverse inclusions are false, so no other canonical order exists.

Is a quasivariety the same as a variety?

No. A quasivariety is defined by implications between conjunctions of equations rather than by equations alone, and is closed under S, P and ultraproducts but not necessarily under H. Fields form a quasivariety-like class that is not a variety. Deciding whether a given quasivariety happens to be a variety is itself a studied and non-trivial question.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.