Connections with Model Theory
The First Two Finite Basis Theorems
Two results giving conditions under which a variety has a finite equational basis, and the general shape of finite basis arguments.
Learning objectives
- State the finite basis problem
- Give the two preliminary finite basis theorems
- Identify the common structure of the arguments
The finite basis problem
A finite set Σ of identities with V = M(Σ) — equivalently, the variety is a basic elementary class in the equational fragment.
Birkhoff's theorem guarantees an equational basis exists for every variety but says nothing about its size. The finite basis problem asks when a finite one exists.
Lyndon produced a seven-element algebra whose variety has no finite basis, and later examples are smaller. So the answer is genuinely conditional, and identifying the right hypotheses is the content of §4.
- Tarski's finite basis problem
- Is it decidable whether a given finite algebra has a finitely based variety?
- Answer
- No — proved undecidable by McKenzie in 1996, after the source
- Consequence
- No general criterion exists; sufficient conditions are the best available
The first theorem
If a variety V of finite type is generated by a finite algebra, has definable principal congruences, and its subdirectly irreducible members are bounded in size, then V has a finite equational basis.
The argument constructs a finite basis explicitly. Identities in enough variables to distinguish the bounded irreducibles suffice, and there are finitely many such identities up to equivalence in a finite type.
The second theorem
Under congruence-permutability together with a finiteness condition on the irreducibles, a finite basis exists.
The permutable case is easier than the general one because principal congruences are described by chains of length one, which gives definable principal congruences immediately.
The common structure
Every finite basis argument has the same three-part shape.
Step 3 fails for infinite types: even in one variable there are infinitely many terms if there are infinitely many operation symbols. Every finite basis theorem assumes a finite similarity type, and modules over infinite rings fall outside the scope for exactly this reason.
| Hypothesis | Role |
|---|---|
| Finite type | Ensures finitely many identities in bounded variables |
| Finitely generated | Gives Jónsson's lemma its force |
| Congruence-distributive | Bounds the irreducibles via Jónsson |
| Definable principal congruences | Makes irreducibility first-order |
| Bounded irreducibles | Bounds the variables needed |
What remains for Baker
The two theorems above assume definable principal congruences, which is a strong hypothesis that lattices and many other congruence-distributive varieties fail. Baker's theorem removes it, replacing DPC by the weaker condition of definable principal subcongruences, and is the culminating result of the section.
Frequently asked questions
Is having a finite basis preserved by subvarieties?
No. A finitely based variety can have subvarieties with no finite basis, and conversely. Finite basedness is not inherited in either direction.
Why is the finite basis problem interesting beyond aesthetics?
Because a finite basis makes the equational theory finitely axiomatised, which is a prerequisite for effective decision procedures and for practical algebraic specification.
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 V.4, book pages 259-265.
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.