Research Frontier and Sourcing
The Seventeen Open Problems: Status Then and Now
Seventeen numbered problems, posed in 1981. Some are settled, some remain open, and for several the honest answer is that the status needs checking against current literature rather than asserting from memory.
- List the seventeen problems as the source states them.
- Group them by the section that poses them.
- Identify the problems whose resolution is well established.
- Identify those that remain open.
- Recognise where the status is uncertain and should be verified.
- Understand why this page separates reporting from asserting.
01The problems as posed
Stated as in the source, grouped by the section that poses them.
| № | § | Problem |
|---|---|---|
| 1 | 1 | For which varieties can we define a commutator? |
| 2 | 1 | Find a description of all A (parallel to II§13.4) such that Z(A) = ∇_A. |
| 3 | 3 | Which locally finite varieties of finite type have a decidable theory? |
| 4 | 3 | For which varieties of finite type is the theory of the finite algebras in the variety decidable? |
| 5 | 3 | Do the finite algebras in any finitely generated arithmetical variety of finite type have a decidable theory? |
| 6 | 3 | Do the finite algebras in any finitely generated congruence-distributive, but not congruence-permutable, variety of finite type have an undecidable theory? |
| 7 | 3 | Is there an algorithm to decide which equations in at most 4 variables hold in modular lattices? |
| 8 | 3 | (as numbered in the source's word-problem discussion) |
| 9 | 3 | Can one derive the Linial–Post theorem from the base-undecidability result on Boolean algebras, or vice versa? |
| 10 | 3 | (Tarski) Is there an algorithm to determine if V(A) has a finitely based equational theory, given that A is a finite algebra of finite type? |
| № | § | Problem |
|---|---|---|
| 11 | 5 | For which varieties does there exist a bound on the size of the directly indecomposable members? |
| 12 | 5 | For which varieties V is every algebra in V a Boolean product of directly indecomposable algebras? Of subdirectly irreducible algebras? Of simple algebras? (Krauss and Clark) |
| 13 | 5 | For which finite rings R with 1 is the variety of unitary left R-modules directly representable? |
| 14 | 8 | (finite basis theorems) |
| 15 | 8 | (finite basis theorems) |
| 16 | 9 | (subdirectly irreducible algebras) |
| 17 | 9 | (subdirectly irreducible algebras) |
These are stated in the source in sections this page summarises rather than reproduces in full. The exact wording is in the source at pp.283–290; the collection does not transcribe it. Problems 14 and 15 concern finite basis theorems and 16 and 17 concern the size and structure of subdirectly irreducible algebras.
02Problem 10: settled
Tarski's problem — whether the finite basis property is decidable for finite algebras of finite type — is the one whose resolution is best established.
McKenzie proved in 1996 that Tarski's finite basis problem is undecidable: there is no algorithm which, given a finite algebra of finite type, decides whether the variety it generates is finitely based. The problem the source records as open has been settled, and settled in the negative.
The result is treated on its own page in this stream, which also covers the positive finite basis theorems obtained since Baker's. The negative resolution does not diminish those — it establishes that no uniform criterion exists, so hypothesis-specific theorems are the only available route.
03Problems 3–6: substantially advanced
The decidability questions have seen major progress, chiefly through tame congruence theory and the structure theory it enabled.
- Tame congruence theory arrivesHobby and McKenzie's 1988 monograph classifies the local behaviour of finite algebras into five types, giving a much finer instrument than the congruence-condition hierarchy of 1981.
- Decidability results followThe classification of decidable locally finite varieties advanced considerably using the type-set machinery, extending the Burris–McKenzie picture the source reports.
- The problems as posed are largely supersededRather than being answered as stated, Problems 3 and 4 were reframed by the new machinery. This is a common fate for problems posed before the right tools exist.
- Status of 5 and 6 specificallyThese are narrow technical questions about finitely generated arithmetical and congruence-distributive varieties. Their current status should be checked against the literature rather than asserted here.
04Problem 7: uncertain status
The four-variable modular lattice question sits precisely in the gap between two known results.
| Variables | Status as at 1981 | Source |
|---|---|---|
| ≤ 3 | decidable | Dedekind's description of the free modular lattice on 3 generators |
| 4 | open — Problem 7 | the gap |
| ≥ 5 | undecidable | Freese 1979 |
The four-variable case is a narrow technical question and its current status is not something to state from memory. A reader needing the answer should check the current lattice theory literature. Stating a confident answer here would be exactly the kind of plausible-looking fabrication this collection's sourcing policy exists to prevent.
05Problems 11–13: structure theory
These problems are more tightly bound to the Chapter IV machinery than the decidability questions and have accordingly attracted a narrower literature. Their status is a matter for current sources.
06Why this page reports rather than asserts
The collection's two-layer policy applies to problem status exactly as it applies to numeric catalogue data.
The most dangerous output here is not an admitted gap but a confidently wrong status line in a table — indistinguishable from a correct one, and repeated by anyone who reads it. That is the same failure mode as a digit substitution in a transcribed numeric table, and it is guarded against the same way.
Frequently asked
Where can I check current status?
The Algebra Universalis journal, the arXiv math.RA and math.LO listings, and the surveys and monographs listed on the sourcing policy page in this stream. For the decidability and finite basis questions specifically, the literature descending from Hobby–McKenzie is the place to start.
Why not simply omit the problems whose status is uncertain?
Because the problems themselves are durable content and are part of what the source records. Omitting them would leave a reader unaware they were ever asked. Recording the problem and marking the status as needing verification gives the reader everything actually known, with the uncertainty visible.
Has the general conviction about decidability and structure held up?
Broadly yes. The pattern the authors identified — that decidable varieties turn out to have structure theorems attached — has continued to hold, and tame congruence theory gave it a much sharper form. The conviction stated in 1981 as a working hypothesis has aged considerably better than the specific open problems.
- 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.