Boolean Constructions and Discriminator Varieties
Semisimple and Directly Representable Varieties
Varieties whose subdirectly irreducible members are all simple, and those whose finite members are all direct products from a fixed finite list.
Learning objectives
- Define semisimple and directly representable varieties
- Identify examples of each
- State the structural consequences
Semisimple varieties
A variety in which every subdirectly irreducible member is simple.
By Birkhoff's theorem, every member of a semisimple variety is a subdirect product of simple algebras — the strongest form of subdirect decomposition.
| Variety | Semisimple? | Subdirect irreducibles |
|---|---|---|
| Boolean algebras | Yes | Only 2, which is simple |
| Distributive lattices | Yes | The two-element chain |
| K-vector spaces | Yes | The one-dimensional space |
| Any discriminator variety | Yes | Simple by the discriminator argument |
| Abelian groups | No | The Prüfer groups are irreducible, not simple |
| Groups | No | Cp2 is irreducible, not simple |
| Lattices | No | Many non-simple irreducibles |
Simple algebras admit no proper quotients, so a subdirect representation into simple factors cannot be refined further. The decomposition is terminal, and classifying the variety reduces to classifying its simple members.
Characterisation attempts
Semisimplicity is not a Mal'cev condition and has no term characterisation in general. What is known relates it to congruence conditions.
- Every discriminator variety is semisimple.
- A congruence-distributive variety need not be semisimple — lattices are a counterexample.
- In congruence-modular varieties, semisimplicity is connected to the commutator vanishing appropriately, a result belonging to the post-1981 commutator theory.
- Semisimplicity is preserved by subvarieties: a subvariety of a semisimple variety is semisimple.
Directly representable varieties
A variety in which every finite member is isomorphic to a direct product of algebras from a fixed finite list of finite algebras.
This is a very strong finiteness condition. It says the finite members are completely classified by a finite amount of data plus multiplicity.
| Variety | Directly representable? | The finite list |
|---|---|---|
| Boolean algebras | Yes | {2} |
| K-vector spaces, K finite | Yes | The one-dimensional space |
| V(A) for primal A | Yes | {A} |
| Abelian groups | No | Infinitely many indecomposable finite abelian groups |
| Distributive lattices | No | Finite distributive lattices are not products of a bounded list |
Every finite Boolean algebra is a power of 2, but not every finite distributive lattice is — the three-element chain is directly indecomposable and not a product of two-element chains. Direct representability requires the indecomposables to be finite in number, and distributive lattices have infinitely many.
The relationship between the two
Directly representable varieties are rare and strongly constrained. The source treats them in §13 as the final and most restrictive class in the chapter's hierarchy of well-behaved varieties.
Chapter IV moves through progressively narrower and better-understood classes: all Boolean algebras, then algebras with Boolean product representations, then discriminator varieties, then quasiprimal and primal cases, and finally the semisimple and directly representable conditions. Each step trades generality for structural completeness.
Frequently asked questions
Does semisimple imply congruence-distributive?
No. Vector spaces over a fixed field are semisimple but congruence-modular rather than congruence-distributive. The two conditions are independent.
Is direct representability decidable for a given finitely generated variety?
The general question is difficult. For congruence-distributive varieties generated by finite algebras, Jónsson's lemma bounds the candidates and makes the question more tractable, but no general decision procedure is claimed in the source.
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 IV.12-13, book pages 207-216.
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.