{ "model": { "rank": 11136, "code": "thing-q64861", "model_id": "vr.tr.transitive-relation", "name": "transitive relation", "purpose": "Let an agent explain transitive relations, relay definitions, examples and related order concepts from mathematics and logic sources, describe the named order types and closures, flag the Wikidata property aliases, and distinguish transitivity from reflexivity, symmetry, quasitransitivity and non-transitive relations such as rock-paper-scissors dominance.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q64861/", "tier": 2, "score": 43, "payload": { "layer": "wikidata", "aliases": [ "strict order", "preorder", "transitive closure", "follows", "followed by", "dense preorder", "weak ordering", "majorization", "well-quasi-ordering", "specialization (pre)order", "Prewellordering", "reflexive transitive closure", "better-quasi-ordering" ], "aliasCount": 13, "merged": 13, "knownIn": 43, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q64861", "url": "https://www.wikidata.org/wiki/Q64861", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T20:00:37Z", "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": 4, "engine": "claude" }, "metaModel": { "id": "THING-Q64861", "registryId": "vr.tr.transitive-relation", "name": "transitive relation", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q64861/", "model": { "registry_id": "vr.tr.transitive-relation", "name": "transitive relation", "purpose": "Let an agent explain transitive relations, relay definitions, examples and related order concepts from mathematics and logic sources, describe the named order types and closures, flag the Wikidata property aliases, and distinguish transitivity from reflexivity, symmetry, quasitransitivity and non-transitive relations such as rock-paper-scissors dominance.", "definition": "A binary relation R on a set such that whenever a R b and b R c, then a R c, as with less than, ancestor of or equality, underpinning preorders, which are reflexive and transitive, strict orders, which are irreflexive and transitive, and dense preorders, while the transitive closure of a relation is the smallest transitive relation containing it; the registry aliases follows and followed by are Wikidata sequence properties, which are not transitive in themselves, and appear misfiled.", "what_it_is_for": "Not applicable; a mathematical property.", "affordances": [ "explain the property", "relay examples", "describe named order types", "distinguish related properties" ], "distinguishing_features": [ "Chaining property", "Basis of orders", "Closure operation", "Aliases include data properties" ], "appearance": "Not a visible object; seen in diagrams as paths implying direct arrows.", "visual_identification": [ "Relation where a R b and b R c imply a R c", "Strict order, preorder, transitive closure, dense preorder, follows, followed by", "Reflexive relations relate each element to itself; symmetric ones go both ways; quasitransitive relations are weaker; follows and followed by are sequence properties" ], "physical_properties": [ { "quantity": "definition", "typical_range": "for all a, b, c: aRb and bRc imply aRc", "unit": "formula", "note": "" }, { "quantity": "Warshall algorithm", "typical_range": "O(n^3)", "unit": "complexity", "note": "transitive closure" }, { "quantity": "registry parents", "typical_range": "endorelation, quasitransitive relation", "unit": "note", "note": "" } ], "families_and_kinds": [ "preorders", "partial orders", "strict orders", "equivalence relations", "dense orders", "transitive closures" ], "related_models": [ { "relation": "is a kind of", "target": "endorelation", "why": "in registry terms" }, { "relation": "underlies", "target": "partial order", "why": "" }, { "relation": "has closure", "target": "transitive closure", "why": "" }, { "relation": "is contrasted with", "target": "intransitivity", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "No regulation" ], "failure_modes_and_hazards": [ "Assuming preferences are always transitive", "Registry aliases from Wikidata properties", "Confusing transitive and quasitransitive" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a transitive relation is.", "rationale": "Science.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a transitive relation, and how does it differ from reflexive, symmetric and quasitransitive relations?", "kind": "definition" }, { "text": "Is the question about maths, logic, databases, preferences or grammar s transitive verbs?", "kind": "boundary" } ] } ] }, { "id": "named", "name": "Named", "description": "Named concepts.", "findings": [ { "id": "named-finding", "name": "Named", "description": "Named.", "questions": [ { "text": "What are strict orders, preorders, dense preorders and transitive closures, and why are follows and followed by different?", "kind": "definition" }, { "text": "Which entry fits the specific concept?", "kind": "action" } ] } ] } ] }, { "id": "mathematics", "name": "Mathematics", "description": "Mathematics.", "rationale": "Science.", "layers": [ { "id": "orders", "name": "Orders", "description": "Order theory.", "findings": [ { "id": "orders-finding", "name": "Orders", "description": "Orders.", "questions": [ { "text": "How do transitive relations build order theory?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "closure", "name": "Closure", "description": "Transitive closure.", "findings": [ { "id": "closure-finding", "name": "Closure", "description": "Closure.", "questions": [ { "text": "How is the transitive closure computed, for example by Warshall s algorithm?", "kind": "provenance" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Applications.", "rationale": "Application.", "layers": [ { "id": "graphs", "name": "Graphs", "description": "Graphs and databases.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "How is transitivity used in graph reachability and database queries?", "kind": "provenance" }, { "text": "Which entry fits reachability?", "kind": "action" } ] } ] }, { "id": "preferences", "name": "Preferences", "description": "Preferences.", "findings": [ { "id": "preferences-finding", "name": "Preferences", "description": "Preferences.", "questions": [ { "text": "Why is transitivity of preferences debated in economics, with evidence attributed?", "kind": "provenance" }, { "text": "Which entry fits rational choice theory?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Logic.", "rationale": "Context.", "layers": [ { "id": "logic", "name": "Logic", "description": "Logic.", "findings": [ { "id": "logic-finding", "name": "Logic", "description": "Logic.", "questions": [ { "text": "How is transitivity expressed in syllogisms and description logics?", "kind": "provenance" }, { "text": "Which entry fits syllogism?", "kind": "action" } ] } ] }, { "id": "nontransitive", "name": "Non-transitive", "description": "Non-transitive examples.", "findings": [ { "id": "nontransitive-finding", "name": "Non-transitive", "description": "Non-transitive.", "questions": [ { "text": "What are examples of non-transitive relations such as nontransitive dice?", "kind": "provenance" }, { "text": "Which entry fits nontransitive dice?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should preorder and strict order be separate primary entries?", "How should mathematics sources be linked?", "The registry aliases follows and followed by should be removed or moved to Wikidata property entries" ], "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 transitive relation, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain transitive relations, relay definitions, examples and related order concepts from mathematics and logic sources, describe the named order types and closures, flag the Wikidata property aliases, and distinguish transitivity from reflexivity, symmetry, quasitransitivity and non-transitive relations such as rock-paper-scissors dominance.", "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: strict order, preorder, transitive closure, follows, followed by, dense preorder, weak ordering, majorization, well-quasi-ordering, specialization (pre)order, Prewellordering, reflexive transitive closure.", "13 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 43 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 transitive relation, 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": [ "strict order", "preorder", "transitive closure", "follows", "followed by", "dense preorder", "weak ordering", "majorization", "well-quasi-ordering", "specialization (pre)order", "Prewellordering", "reflexive transitive closure" ] } ], "standing": "Described in 43 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 13 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 transitive relation from another, and a transitive relation 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 transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation 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 transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation 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 transitive relation between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a transitive relation 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 transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation 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 transitive relation 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 transitive relation 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 transitive relation 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 transitive relation?", "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 transitive relation?", "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 transitive relation 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 transitive relation and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a transitive relation, 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 transitive relation, 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 transitive relation 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 transitive relation is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a transitive relation, 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 transitive relation, 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 transitive relation 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 } } } }