{ "model": { "rank": 5016, "code": "thing-q164307", "model_id": "vr.tr.binary-operation", "name": "binary operation", "purpose": "Let an agent handle binary operations by definition, properties, examples and role in algebraic structures.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q164307/", "tier": 2, "score": 70, "payload": { "layer": "wikidata", "aliases": [ "super-logarithm", "function composition", "interior product", "graph product", "internal binary operation", "Poisson bracket", "law of composition", "cross product", "modulo operation", "tetration", "mediant", "pentation", "external binary operation", "binary function that is not an internal binary operation", "product in algebraic topology", "knot sum", "internal binary function", "group operation", "function iteration", "partial application" ], "aliasCount": 20, "merged": 21, "knownIn": 70, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q164307", "url": "https://www.wikidata.org/wiki/Q164307", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T10:13:09Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q164307", "registryId": "vr.tr.binary-operation", "name": "binary operation", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q164307/", "model": { "registry_id": "vr.tr.binary-operation", "name": "binary operation", "purpose": "Let an agent handle binary operations by definition, properties, examples and role in algebraic structures.", "definition": "An operation that combines two elements of a set to produce an element, such as addition, multiplication, composition of functions or the Poisson bracket; properties include associativity, commutativity, identity and inverses.", "what_it_is_for": "Defining algebraic structures such as groups, rings and fields.", "affordances": [ "check properties of an operation", "give examples and counterexamples", "identify algebraic structures", "use operation tables" ], "distinguishing_features": [ "Takes two inputs", "Closed or partial", "Properties define structures", "Everywhere in mathematics and computing" ], "appearance": "Written with symbols such as +, x, composition circles or brackets.", "visual_identification": [ "Notation a * b", "Cayley tables", "Unary and ternary operations take one or three inputs" ], "physical_properties": [], "families_and_kinds": [ "arithmetic operations", "function composition", "logical operations", "products in algebra", "partial binary operations" ], "related_models": [ { "relation": "is a kind of", "target": "mathematical operation and binary function", "why": "category" }, { "relation": "defines", "target": "group and ring", "why": "structures" }, { "relation": "is studied in", "target": "abstract algebra", "why": "field" }, { "relation": "is related to", "target": "logical relation", "why": "logic" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 notation" ], "failure_modes_and_hazards": [ "Assuming commutativity", "Ignoring closure" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "Operation.", "rationale": "Definitions must be precise.", "layers": [ { "id": "set", "name": "Set", "description": "Domain.", "findings": [ { "id": "set-finding", "name": "Set", "description": "Domain.", "questions": [ { "text": "On which set is the operation defined?", "kind": "definition" }, { "text": "Is it closed on that set?", "kind": "boundary" } ] } ] }, { "id": "notation", "name": "Notation", "description": "Symbols.", "findings": [ { "id": "notation-finding", "name": "Notation", "description": "Notation.", "questions": [ { "text": "How is the operation written?", "kind": "definition" }, { "text": "Is the notation standard?", "kind": "provenance" } ] } ] } ] }, { "id": "properties", "name": "Properties", "description": "Laws.", "rationale": "Properties define structure.", "layers": [ { "id": "associativity", "name": "Associativity", "description": "Grouping.", "findings": [ { "id": "associativity-finding", "name": "Associativity", "description": "Associativity and commutativity.", "questions": [ { "text": "Is the operation associative and commutative?", "kind": "boundary" }, { "text": "Can a counterexample be given?", "kind": "action" } ] } ] }, { "id": "identity", "name": "Identity", "description": "Identity and inverses.", "findings": [ { "id": "identity-finding", "name": "Identity", "description": "Identity and inverses.", "questions": [ { "text": "Is there an identity element and do inverses exist?", "kind": "boundary" }, { "text": "What are they?", "kind": "definition" } ] } ] } ] }, { "id": "structures", "name": "Structures", "description": "Algebra.", "rationale": "Operations build structures.", "layers": [ { "id": "structure", "name": "Structure", "description": "Group, ring.", "findings": [ { "id": "structure-finding", "name": "Structure", "description": "Algebraic structure.", "questions": [ { "text": "Which algebraic structure does the set with this operation form?", "kind": "definition" }, { "text": "Which axioms fail if any?", "kind": "boundary" } ] } ] }, { "id": "tables", "name": "Tables", "description": "Cayley tables.", "findings": [ { "id": "tables-finding", "name": "Tables", "description": "Operation tables.", "questions": [ { "text": "What does the Cayley table look like for this finite set?", "kind": "action" }, { "text": "What patterns does it show?", "kind": "definition" } ] } ] } ] }, { "id": "computing", "name": "Computing", "description": "Programming.", "rationale": "Operations appear in code.", "layers": [ { "id": "code", "name": "Code", "description": "Operators.", "findings": [ { "id": "code-finding", "name": "Code", "description": "Programming operators.", "questions": [ { "text": "How is the operation implemented in this language?", "kind": "action" }, { "text": "Are there overflow or precision issues?", "kind": "boundary" } ] } ] }, { "id": "teaching", "name": "Teaching", "description": "Learning.", "findings": [ { "id": "teaching-finding", "name": "Teaching", "description": "Teaching.", "questions": [ { "text": "Which examples help learners understand binary operations?", "kind": "action" }, { "text": "Which misconceptions arise?", "kind": "boundary" } ] } ] } ] } ] }, "openQuestions": [ "Should properties be separate entries?", "How should algebraic structures be linked?", "How should partial operations be handled?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a binary operation, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent handle binary operations by definition, properties, examples and role in algebraic structures.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: super-logarithm, function composition, interior product, graph product, internal binary operation, Poisson bracket, law of composition, cross product, modulo operation, tetration, mediant, pentation.", "21 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 70 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a binary operation, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "super-logarithm", "function composition", "interior product", "graph product", "internal binary operation", "Poisson bracket", "law of composition", "cross product", "modulo operation", "tetration", "mediant", "pentation" ] } ], "standing": "Described in 70 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 21 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one binary operation from another, and a binary operation from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a binary operation sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a binary operation from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a binary operation passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a binary operation between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a binary operation, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a binary operation is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What binary operation is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of binary operation are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is binary operation operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of binary operation and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of binary operation?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as binary operation?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a binary operation came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a binary operation and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a binary operation is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a binary operation, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a binary operation, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a binary operation is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }