Recent Developments and Resources
The Seventeen Open Problems: a Status Register
A status register for the open problems stated in the source's closing chapter, reporting what is known rather than asserting resolutions.
Learning objectives
- Understand the scope and limits of this register
- Locate the problems by theme
- Treat status claims with appropriate caution
Material on this page extends past the 1981 text and its Millennium re-typesetting. Statements here are attributed to later literature, not to Burris and Sankappanavar. Where the status of a question is unsettled, this page says so rather than resolving it.
Scope and caution
The source's Recent Developments chapter states 17 numbered open problems, grouped across its nine sections. This page reports what is known about them by theme. It does not claim to resolve any problem, and where the status is uncertain it says so rather than guessing. Readers should verify current status against the literature; several of these have seen substantial work since.
The problems were stated around 1981. Some have been resolved, some remain open, and for some the precise formulation in the source differs from the version that later literature addresses. Distinguishing these cases reliably requires access to the current literature.
Themes and status
| Theme | Source section | Status |
|---|---|---|
| Commutator and centre | RD §1 | Substantially developed by Freese and McKenzie; the abelian-algebras-are-modules theorem is established |
| Classification of varieties | RD §2 | Tame congruence theory (Hobby–McKenzie 1988) provides a framework; specific classification questions remain |
| Decidability | RD §3 | Locally finite decidable varieties characterised by McKenzie and Valeriote (1989); Tarski's finite basis problem shown undecidable by McKenzie (1996) |
| Boolean constructions | RD §4 | Natural duality theory developed; specific representation questions vary |
| Structure theory | RD §5 | Advanced substantially; the finite lattice representation problem remains open |
| Applications to computer science | RD §6 | CSP dichotomy resolved 2017; other questions ongoing |
| Applications to model theory | RD §7 | Ongoing; no single resolution |
| Finite basis theorems | RD §8 | Baker, McKenzie and Willard theorems established; Park's conjecture open |
| Subdirectly irreducible algebras | RD §9 | Residual smallness characterised for congruence-modular varieties |
Problems believed still open
- The finite lattice representation problem. Is every finite lattice the congruence lattice of a finite algebra? Widely regarded as open and as one of the central problems of the subject.
- Park's conjecture. Is every finitely generated residually finite variety of finite type finitely based? Proved under additional hypotheses by Willard; open in general.
- Complete classification of finite simple algebras up to term equivalence. Advanced but not complete.
Reporting a problem as open requires confidence that no resolution has appeared. For the majority of the 17, that confidence is not available here without checking the current literature, so the register reports themes and known developments rather than a problem-by-problem verdict.
How to use this register
For current status on any of these, the standard references are McKenzie, McNulty and Taylor, Algebras, Lattices, Varieties (1987, reissued 2018), Hobby and McKenzie, The Structure of Finite Algebras (1988), and Freese and McKenzie, Commutator Theory for Congruence Modular Varieties (1987). Each contains problem lists updating the source's.
Frequently asked questions
Why not give a definitive status for each of the 17 problems?
Because doing so accurately requires checking each against the current literature, and stating a confident resolution that turns out to be wrong would be worse than reporting uncertainty. The themes and major developments are reported; individual verdicts are not.
Have any of the problems been shown to be independent of ZFC?
Not to the knowledge reflected here. Some questions in the vicinity — particularly about cardinal invariants of congruence lattices — do interact with set theory, but no claim is made about the specific problems 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 RD, book pages 283-290.
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.