{ "model": { "rank": 11841, "code": "thing-q381060", "model_id": "vr.tr.partition-of-a-set", "name": "partition of a set", "purpose": "Let an agent explain set partitions, relay definitions, counting, equivalence relations and applications from mathematics sources, describe the named related partitions, and distinguish set partitions from integer partitions, covers that may overlap, subsets and partitions in computing such as disk partitions.", "family": "Thing Registry", "category": "Activities and processes", "status": "research-draft", "kind": "thing", "plane": "ACT", "domain": "ACT.ACT", "industry": "", "version": "", "url": "/models/thing/q381060/", "tier": 2, "score": 41, "payload": { "layer": "wikidata", "aliases": [ "trichotomy", "Arrangement of pseudolines", "cut", "arrangement of lines", "Voronoi diagram", "Dedekind cut", "minimum cut", "maximum cut", "Discrete Voronoi diagram", "Weighted Voronoi diagram" ], "aliasCount": 10, "merged": 10, "knownIn": 41, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q381060", "url": "https://www.wikidata.org/wiki/Q381060", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-13T22:37:14Z", "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-Q381060", "registryId": "vr.tr.partition-of-a-set", "name": "partition of a set", "version": "0.2.0-wave.4", "entryKind": "thing", "family": "Thing Registry", "domain": [ "ACT.ACT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q381060/", "model": { "registry_id": "vr.tr.partition-of-a-set", "name": "partition of a set", "purpose": "Let an agent explain set partitions, relay definitions, counting, equivalence relations and applications from mathematics sources, describe the named related partitions, and distinguish set partitions from integer partitions, covers that may overlap, subsets and partitions in computing such as disk partitions.", "definition": "In mathematics, a grouping of the elements of a set into non-empty subsets called blocks such that every element belongs to exactly one block, corresponding one-to-one with equivalence relations, counted by Bell numbers for all partitions and by Stirling numbers of the second kind for a fixed number of blocks; related constructions the registry aliases name include a cut, a partition of graph vertices into two sets, a Dedekind cut, which partitions the rationals to define real numbers, trichotomy, the division of numbers into less than, equal and greater, arrangements of lines and pseudolines, which partition the plane into regions, and Voronoi diagrams, which partition space by nearest site.", "what_it_is_for": "Dividing sets into disjoint parts.", "affordances": [ "explain definition and equivalence relations", "relay counting with Bell and Stirling numbers", "describe named related partitions", "distinguish related concepts" ], "distinguishing_features": [ "Disjoint blocks", "Complete coverage", "Equivalence relation link", "Combinatorial counts" ], "appearance": "Not a physical object; drawn as disjoint groups of elements.", "visual_identification": [ "Division of a set into disjoint non-empty blocks", "Trichotomy, arrangement of pseudolines, cut, arrangement of lines, Voronoi diagram, Dedekind cut", "Integer partitions split numbers; covers may overlap; subsets are single parts; disk partitions divide storage" ], "physical_properties": [ { "quantity": "Bell numbers", "typical_range": "1, 1, 2, 5, 15, 52", "unit": "sequence", "note": "partitions of sets of size 0-5" }, { "quantity": "Dedekind cuts", "typical_range": "1872", "unit": "year", "note": "" }, { "quantity": "Voronoi diagram", "typical_range": "Voronoi 1908", "unit": "note", "note": "earlier uses by Descartes and Dirichlet" } ], "families_and_kinds": [ "set partitions", "graph cuts", "Dedekind cuts", "geometric partitions such as line arrangements and Voronoi diagrams", "noncrossing partitions" ], "related_models": [ { "relation": "is a kind of", "target": "set system", "why": "in registry terms" }, { "relation": "corresponds to", "target": "equivalence relation", "why": "" }, { "relation": "is counted by", "target": "Bell number", "why": "" }, { "relation": "is contrasted with", "target": "integer partition", "why": "" } ], "identifiers": [], "standards_and_regulation": [ "No regulation" ], "failure_modes_and_hazards": [ "Confusing set and integer partitions", "Allowing empty blocks", "Treating related geometric partitions as synonyms" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "understand", "name": "Understand", "description": "What a set partition is.", "rationale": "Definition.", "layers": [ { "id": "definition", "name": "Definition", "description": "Definition.", "findings": [ { "id": "definition-finding", "name": "Definition", "description": "Definition.", "questions": [ { "text": "What is a partition of a set, and how does it differ from integer partitions, covers, subsets and disk partitions?", "kind": "definition" }, { "text": "Is the question about combinatorics, geometry, analysis or computing?", "kind": "boundary" } ] } ] }, { "id": "related", "name": "Related", "description": "Named related partitions.", "findings": [ { "id": "related-finding", "name": "Related", "description": "Related.", "questions": [ { "text": "What are cuts, Dedekind cuts, trichotomy, line and pseudoline arrangements and Voronoi diagrams?", "kind": "definition" }, { "text": "Which entry fits the specific concept?", "kind": "action" } ] } ] } ] }, { "id": "theory", "name": "Theory", "description": "Theory.", "rationale": "Science.", "layers": [ { "id": "equivalence", "name": "Equivalence", "description": "Equivalence relations.", "findings": [ { "id": "equivalence-finding", "name": "Equivalence", "description": "Equivalence.", "questions": [ { "text": "Why do partitions correspond to equivalence relations?", "kind": "provenance" }, { "text": "Which references are standard?", "kind": "provenance" } ] } ] }, { "id": "counting", "name": "Counting", "description": "Bell and Stirling numbers.", "findings": [ { "id": "counting-finding", "name": "Counting", "description": "Counting.", "questions": [ { "text": "How do Bell numbers and Stirling numbers count partitions?", "kind": "measurement" }, { "text": "Which sources are cited?", "kind": "provenance" } ] } ] } ] }, { "id": "applications", "name": "Applications", "description": "Applications.", "rationale": "Science.", "layers": [ { "id": "geometry", "name": "Geometry", "description": "Geometric partitions.", "findings": [ { "id": "geometry-finding", "name": "Geometry", "description": "Geometry.", "questions": [ { "text": "How are Voronoi diagrams used in geography, biology and computing?", "kind": "provenance" }, { "text": "Which entry fits Voronoi diagram?", "kind": "action" } ] } ] }, { "id": "graphs", "name": "Graphs", "description": "Graph cuts.", "findings": [ { "id": "graphs-finding", "name": "Graphs", "description": "Graphs.", "questions": [ { "text": "What are cuts in graphs, and how does the max-flow min-cut theorem use them?", "kind": "provenance" }, { "text": "Which entry fits max-flow min-cut theorem?", "kind": "action" } ] } ] } ] }, { "id": "context", "name": "Context", "description": "Foundations.", "rationale": "Context.", "layers": [ { "id": "dedekind", "name": "Dedekind", "description": "Dedekind cuts.", "findings": [ { "id": "dedekind-finding", "name": "Dedekind", "description": "Dedekind.", "questions": [ { "text": "How did Dedekind use cuts to construct the real numbers?", "kind": "provenance" }, { "text": "Which entry fits Richard Dedekind?", "kind": "action" } ] } ] }, { "id": "lattice", "name": "Lattice", "description": "Partition lattice.", "findings": [ { "id": "lattice-finding", "name": "Lattice", "description": "Lattice.", "questions": [ { "text": "What is the lattice of partitions ordered by refinement?", "kind": "provenance" }, { "text": "Which entry fits partition lattice?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should Voronoi diagram and Dedekind cut be moved to their own entries?", "How should combinatorics references be linked?", "Should trichotomy be moved to order theory?" ], "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 partition of a set, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain set partitions, relay definitions, counting, equivalence relations and applications from mathematics sources, describe the named related partitions, and distinguish set partitions from integer partitions, covers that may overlap, subsets and partitions in computing such as disk partitions.", "characteristics": { "substance": "activity", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: trichotomy, Arrangement of pseudolines, cut, arrangement of lines, Voronoi diagram, Dedekind cut, minimum cut, maximum cut, Discrete Voronoi diagram, Weighted Voronoi diagram.", "10 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 41 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 partition of a set, and on what evidence are they told apart?" ], "whatItIsMadeOf": "something that happens over time", "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": [ "trichotomy", "Arrangement of pseudolines", "cut", "arrangement of lines", "Voronoi diagram", "Dedekind cut", "minimum cut", "maximum cut", "Discrete Voronoi diagram", "Weighted Voronoi diagram" ] } ], "standing": "Described in 41 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 10 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 partition of a set from another, and a partition of a set 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 partition of a set, 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 partition of a set, 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 partition of a set 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 partition of a set, 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 partition of a set, 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 partition of a set 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 partition of a set 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 partition of a set, 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 partition of a set, 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 partition of a set 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 partition of a set from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a partition of a set, 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 partition of a set, 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 partition of a set 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 partition of a set 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 partition of a set between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a partition of a set, 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 partition of a set, 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 partition of a set 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 partition of a set, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a partition of a set 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 partition of a set, 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 partition of a set, 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 partition of a set is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "process-and-outcome", "name": "Process, inputs and outcome", "description": "How a partition of a set proceeds, what it needs and what it leaves behind.", "rationale": "An activity is known by its steps and its results, and both have to be recordable while it is still running.", "layers": [ { "id": "steps-and-sequence", "name": "Steps and sequence", "description": "The steps of a partition of a set, their order and what may run in parallel.", "findings": [ { "id": "steps-and-preconditions", "name": "Steps, preconditions and completion", "description": "What has to be true before each step of a partition of a set and what marks it done.", "questions": [ { "id": "steps-and-preconditions-q01", "text": "What identifies and describes a step of a partition of a set, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "steps-and-preconditions-q02", "text": "Who or what asserted this about a step of a partition of a set, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "steps-and-preconditions-q03", "text": "What may an agent decide or do once a step of a partition of a set is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "inputs-and-results", "name": "Inputs, resources and results", "description": "What a partition of a set consumes and what it produces.", "findings": [ { "id": "inputs-and-outputs", "name": "Inputs, outputs and side effects", "description": "The resources a partition of a set takes and the results it leaves, wanted or not.", "questions": [ { "id": "inputs-and-outputs-q01", "text": "What identifies and describes the inputs and results of a partition of a set, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "inputs-and-outputs-q02", "text": "Who or what asserted this about the inputs and results of a partition of a set, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "inputs-and-outputs-q03", "text": "What may an agent decide or do once the inputs and results of a partition of a set 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 partition of a set 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 partition of a set 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 partition of a set 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 partition of a set 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 partition of a set?", "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 partition of a set?", "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 partition of a set 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 partition of a set and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a partition of a set, 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 partition of a set, 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 partition of a set 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 partition of a set is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a partition of a set, 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 partition of a set, 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 partition of a set is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 6, "layers": 12, "findings": 14, "questions": 38 } } } }