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.
Engineering · Mathematics11 min readKV-MATH-0258
Learning objectives
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.
Problems 1–10
№
§
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?
Problems 11–17
№
§
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)
NoteProblems 8, 14–17
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.
Key resultProblem 10 was answered negatively
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 arrives
Hobby 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 follow
The 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 superseded
Rather 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 specifically
These 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.
What the source establishes around Problem 7
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
CautionThis collection does not assert a resolution for Problem 7
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
Problem 12
Krauss and Clark's question
Which varieties admit Boolean product representations by indecomposable, subdirectly irreducible or simple algebras. Substantial work followed; the discriminator case was already settled by Bulman-Fleming, Keimel and Werner.
Problems 11 and 13
Bounds and module varieties
Bounds on directly indecomposable members, and which finite rings give directly representable module varieties. Both connect to the (discriminator) ⊗ (modular Abelian) classification.
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.
Durable
The problems themselves
What was asked, by whom, and why it mattered. This does not change and is safely recorded here.
Volatile
Current status
Which problems are open shifts as work is published. A confident list would age silently and would look authoritative while being wrong.
The rule applied
State what is well established, flag the rest
Problem 10's negative resolution is well established and stated. Problem 7's status is not, and is flagged rather than guessed.
Key resultThe failure mode being avoided
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.
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.
Handbook application: from concept to controlled practice
Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review The Seventeen Open Problems: Status Then and Now. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.
The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.
Treat The Seventeen Open Problems: Status Then and Now as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—problems, status, problem, open, seventeen—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.
For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.
A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.
Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.
Step-by-step operating method
Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.
Worked-example protocol
Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.
Stage
Record
Quality check
Input
Objects, domain, notation, assumptions
Every symbol is defined
Method
Permitted operation or cited result at each step
All hypotheses hold
Output
Exact result and representation
Correct type, domain and form
Verification
Substitution, invariant or alternative derivation
Independent agreement
Boundary test
Zero, identity, degenerate or failed hypothesis
Scope is understood
Common failure modes and recovery actions
1. Watch for
Using a theorem without checking every hypothesis.
Recovery: Return to the governing definition or requirement and restate the decision in one sentence.
2. Watch for
Treating a suggestive example as a proof of the general case.
Recovery: Separate evidence from assumption, assign an owner and set a date for validation.
3. Watch for
Changing notation or conventions part-way through an argument.
Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.
4. Watch for
Hiding a division-by-zero, convergence, finiteness or commutativity assumption.
Recovery: Record the consequence, decision and rationale, then update the controlled baseline.
5. Watch for
Reporting a computed result without a residual, substitution or structural check.
Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.
Review checklist
Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?
Questions for deeper application
What is the most important distinction a practitioner must preserve when applying The Seventeen Open Problems: Status Then and Now?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about problems would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
The Stacks Project — table of contents — The Stacks Project. Used for commutative algebra, homological algebra, modules and derived categories. Accessed 2026-08-13.
