Research Frontier and Sourcing

Universal Algebra: Computation and Sources

Durable method belongs in a knowledge base. Computed values, enumeration counts and research status do not — they belong to the live tools and databases that maintain them.

Engineering · Mathematics5 min readKV-MATH-0262

Learning objectives

State the two-layer policy and the reasoning behind it.
Identify the categories of data this collection does not transcribe.
Explain why silent corruption is the primary risk.
Locate the current computational tools for universal algebra.
Locate the current literature and reference works.
Apply the policy when extending the collection.

01The policy

This collection carries durable engineering and mathematical method. It does not transcribe computed values, enumeration counts or the current status of research questions.

In the collection

Method

Definitions, theorems, proof strategies, procedures, worked criteria, structural relationships. These do not change and are safe to write down.

Routed elsewhere

Values and status

Design enumeration counts, MOLS bounds, free spectra for named varieties, which open problems are currently open. These change and are routed to maintained sources.

The split is not a limitation of effort but a deliberate design decision, and it is applied consistently across every KEVOS collection.

02Why: silent corruption is the primary risk

The reasoning runs in order of weight.

Silent corruption
Transcription of dense numeric tables produces digit substitutions, dropped decimals and collapsed columns. The output looks correct and is not — far more dangerous than an obvious failure, because nothing signals the error.
Shelf life
Enumeration counts and bounds are the output of computer searches that are extended as computing power grows. A figure correct in 1981 may be superseded, and a page carrying it ages silently.
Research status changes
The seventeen open problems illustrate the point directly: several are settled. A confident status table would be wrong in exactly the way that is hardest to detect.
Licensing
The source is copyrighted, with the Millennium Edition permitting personal copying only. Verbatim extraction is not available regardless of the other reasons.

Key resultThe failure mode being designed against

A plausible-looking fabricated value is worse than an admitted gap. A gap prompts the reader to look it up; a wrong figure gets used. Every page in this collection that touches computed data states the durable facts and routes the numbers.

03What this collection does not transcribe

Categories routed rather than recorded
Category	Appears in	Routed to
Steiner triple system enumeration counts	Applied stream	design theory handbooks and databases
Latin square and MOLS counts, N(n) bounds	Applied stream	combinatorial design references
Free spectra for named varieties	Equational stream	UACalc, computer algebra systems
Explicit equational bases for finite algebras	Model-Theoretic stream	UACalc, automated provers
Current status of the seventeen problems	Frontier stream	current literature
Type sets of specific finite algebras	Frontier stream	UACalc
Complexity classifications of specific CSP templates	Frontier stream	current literature

In each case the page states what is durable — the admissibility condition, the bound N(n) ≤ n − 1, the criterion for tractability — and directs the reader elsewhere for the value.

04Computational tools

UACalc

Universal Algebra Calculator

The primary tool for the subject. Computes congruence lattices, subalgebra lattices, free algebras, Mal'cev conditions, tame congruence types and polymorphism clones for finite algebras. Developed by Freese and Kiss.

GAP

Groups, Algorithms, Programming

General computational discrete algebra. Strong for groups and semigroups, with packages covering related algebraic structures.

Prover9 and Mace4

Automated reasoning

Prover9 searches for equational and first-order proofs; Mace4 searches for finite counterexamples. The standard pair for equational questions.

Waldmeister

Equational theorem proving

Specialised for unit equational logic via Knuth–Bendix completion. Effective where the theory admits a confluent terminating rewriting system.

Mace4 and finite model finders

Counterexample search

For refuting a conjectured identity or exhibiting an algebra with prescribed properties.

Computer algebra systems

Magma, Macaulay2, Singular

General-purpose where the question crosses into commutative algebra or representation theory.

NoteTool availability changes

Names, maintenance status and capabilities of software shift. This page names the tools that were standard in the subject and does not record version numbers, download locations or feature lists, which would date immediately. Check current documentation before relying on a specific capability.

05Literature and reference works

Where to read further
Work	Covers
Burris and Sankappanavar, A Course in Universal Algebra	the source of this collection; the Millennium Edition is author-authorised for personal copying
McKenzie, McNulty and Taylor, Algebras, Lattices, Varieties	the comprehensive treatment; substantially broader than the source
Hobby and McKenzie, The Structure of Finite Algebras	tame congruence theory, 1988
Grätzer, Universal Algebra	the classical reference, with appendices surveying developments
Grätzer, Lattice Theory	the lattice-theoretic background of Chapter I
Chang and Keisler, Model Theory	the model theory of Chapter V; the source's notational reference
Algebra Universalis	the specialist journal for the subject
arXiv math.RA and math.LO	current preprints

For the applied material, the standard combinatorial design references cover Steiner systems and Latin squares, and the automata and formal language literature covers the Eilenberg correspondence and its descendants.

06Applying the policy when extending

ProcedureThe test to apply before writing a value into a page

in: a candidate fact → out: record it, or state-and-route

ask: is this a definition, theorem, procedure or structural relationship?
if yes → write it; it is durable method
ask: is this the output of a computation or search?
if yes → state the durable facts around it, route the value
ask: is this the current status of a research question?
if yes → state what the source says, flag status, route
ask: would a reader be harmed by relying on this if it were stale?
if yes → route rather than record
ask: is this verbatim from a copyrighted source?
if yes → do not reproduce; write original explanation from the topic map

The test is deliberately conservative. The cost of routing a value that would have been fine is a small inconvenience; the cost of recording one that goes stale is an authoritative-looking error with no signal attached.

Every page in this collection is original KEVOS explanatory writing produced from the topic map of the cited works. Nothing is extracted, transcribed or paraphrased closely from the source, which is both a licensing requirement and, given the corruption risk, the safer engineering choice.

Frequently asked

Why not just transcribe the tables and add a date stamp?

Because a date stamp does not prevent use — readers take the value and the stamp goes unread. It also does not address transcription error, which is present from the moment of writing rather than developing over time. Routing to a maintained source addresses both at once.

Does this policy limit what the collection can answer?

For questions of method, no — the collection covers the durable content of the source in full. For questions of the form 'how many Steiner triple systems of order 19 are there', yes deliberately: the collection explains what the question means and where the answer is maintained, rather than supplying a figure it cannot keep current.

How does this apply to the post-source pages?

The same way. Those pages record what is well established — McKenzie's 1996 resolution, the 2017 CSP dichotomy, the five tame congruence types — and decline to assert status for questions where certainty is lacking, such as Problem 7. The three post-source pages are flagged as such in their metadata and in the README so their provenance is never in doubt.

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.