Boolean Constructions and Discriminator Varieties

Directly Representable Varieties

A directly representable variety is built from finitely many finite algebras by direct products alone. McKenzie proved they are congruence-permutable, and their indecomposables are modular Abelian or functionally complete.

Engineering · Mathematics4 min readKV-MATH-0246

Learning objectives

Define direct representability.
State McKenzie's congruence-permutability theorem.
Describe the directly indecomposable members.
Relate the class to modular Abelian and discriminator varieties.
Explain the role in the decidability classification.
Identify the open problems the source records.

01The definition

A variety V is directly representable when there is a finite set of finite algebras such that every finite member of V is isomorphic to a direct product of members of that set.

V directly representable ⟺ ∃ finite K of finite algebras with every finite A ∈ V isomorphic to a direct product from K

Direct products only — no subalgebras, no quotients, no subdirect products. This is a much stronger demand than being generated by a finite set.

Key resultWhy the condition is strong

Most varieties generated by a finite algebra are not directly representable. Being closed under H and S means members typically arise that are not direct products of the generators at all. Direct representability asks that the direct product operation alone suffice for the finite members, which is a severe restriction.

02McKenzie's theorem

The main structural result is that direct representability forces congruence-permutability.

Key resultMcKenzie's theorem

Every directly representable variety is congruence-permutable. Moreover, in a directly representable variety every directly indecomposable algebra is either modular Abelian or functionally complete.

Permutability is derived, not assumed
The hypothesis is combinatorial — finite members factor as direct products — and the conclusion is a Mal'cev condition. That a counting-flavoured hypothesis yields a term condition is the striking part.
Hence modularity
Congruence-permutable implies congruence-modular, so the commutator theory and the centre are available for directly representable varieties.
The dichotomy on indecomposables
Each directly indecomposable member is modular Abelian — polynomially a module — or functionally complete, with every operation a polynomial operation. Two extremes and nothing between.
Structural reading
The variety is assembled from module-like pieces and discriminator-like pieces. This is the local form of the global decomposition below.

03The two kinds of indecomposable

Modular Abelian

Z(A) = ∇

Polynomially equivalent to a module over a ring. All the structure is linear; the centre is everything and the commutator is trivial.

Functionally complete

Z(A) = Δ, simple

Every operation is a polynomial operation. No linear structure at all; the centre is trivial. Discriminator-like behaviour.

The dichotomy is between maximal and minimal centre, with nothing intermediate permitted. That is characteristic of the commutator theory: the centre tends to be all or nothing in well-behaved settings, and the interesting varieties are those where both kinds coexist.

04The global decomposition

The local dichotomy has a global counterpart, and it is the organising result of the classification programme the source describes.

V = V₁ ∨ V₂ with V₁ a discriminator variety and V₂ modular Abelian
written V = (discriminator) ⊗ (modular Abelian)

V is congruence-modular and is the join of the two subvarieties. Every algebra in V decomposes uniquely, up to isomorphism, as a product of one algebra from each.

Where the decomposition appears
Result	Statement
Burris and McKenzie, 1981	A decidable locally finite congruence-modular variety must be of this form.
Same	There is an algorithm deciding, for finite K of finite algebras of finite type, whether V(K) is of this form.
Same	If so, one can construct a finite ring R with 1 such that V(K) is decidable iff the variety of unitary left R-modules is.
Boolean representability	If V = IΓᵃ(K) for finitely many finite algebras, then V is of this form.

So the decomposition is simultaneously the answer to a decidability question and to a representability question. That coincidence is what made this class the focus of the research the source reports on.

05Why direct representability was studied

Structure theory

Complete description available

A directly representable variety is understood: finitely many finite building blocks, combined by direct products, with a classification of the blocks.

Decidability

Tracks the classification

The decidability results for locally finite congruence-modular varieties are stated in terms of the same decomposition.

Boolean constructions

The limit of the method

Directly representable varieties mark how far Boolean-style representation can be pushed. Beyond them the method does not reach.

The source presents this as the endpoint of Chapter IV: having built Boolean powers, Boolean products, discriminator varieties and the primality hierarchy, directly representable varieties are where the accumulated machinery gives a complete answer.

06Open problems recorded in the source

The source's closing survey lists three problems bearing directly on this material.

Problem 11
Bounded indecomposables
For which varieties does there exist a bound on the size of the directly indecomposable members?
Problem 12
Boolean product representations
For which varieties is every algebra a Boolean product of directly indecomposable algebras? Of subdirectly irreducible algebras? Of simple algebras? Posed by Krauss and Clark.
Problem 13
Directly representable module varieties
For which finite rings R with 1 is the variety of unitary left R-modules directly representable?

NoteStatus as at the source

All three were open when the source was written in 1981. The Research Frontier stream reports on the seventeen problems as a set and marks which have since been settled — material that necessarily postdates the text and is flagged as such.

Frequently asked

Is every finitely generated variety directly representable?

No, and most are not. Direct representability requires the finite members to be direct products of a fixed finite list, which fails as soon as H and S produce finite algebras outside that closure. It is a much stronger condition than finite generation.

Does direct representability imply congruence-distributivity?

No — it implies permutability, hence modularity, but not distributivity. Module varieties are directly representable in favourable cases and are permutable without being distributive. The discriminator half of the decomposition is distributive; the modular Abelian half is not.

What does the ⊗ notation mean exactly?

It denotes that the variety is congruence-modular and is the join of a discriminator subvariety and a modular Abelian subvariety, with every member factoring uniquely as a product of one algebra from each. It is not a tensor product in any standard sense — the notation is the source's shorthand for this specific decomposition.

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.