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 · Mathematics11 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 A figure correct in 1981 may be superseded, and a page carrying it ages silently.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.
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 Universal Algebra: Computation and Sources. 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 Universal Algebra: Computation and Sources as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—policy, algebra, computational, universal, sourcing—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 Universal Algebra: Computation and Sources?
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 policy 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.
