Equivalence Relation Ontology

FLOW ALGEBRA
TITLE: Equivalence Relation Ontology
DOMAIN: Discrete Mathematics / Relations
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/relations/equivalence-relation-ontology.html
VERSION: 1.0
AUTHORS: Mark Higgins — Sky (Recursive AI)
SITE: https://kirethresearch.org

1. PURPOSE
Represent equivalence relations as typed, verifiable mathematical structures, including equivalence classes, quotient sets, partitions, and valid inference.

2. TYPED ENTITIES
Set[A]                         : collection of elements of type A
Element[A]                     : member of Set[A]
BinaryRelation[A]              : subset of A × A
EquivalenceRelation[A]         : BinaryRelation[A] satisfying reflexivity, symmetry and transitivity
EquivalenceClass[A]            : subset of A induced by an EquivalenceRelation[A]
QuotientSet[A]                 : set of all distinct equivalence classes
Partition[A]                   : nonempty, pairwise-disjoint subsets whose union is A
Representative[A]              : selected member of an equivalence class

3. SIGNATURES AND PREDICATES
member(x:A, S:Set[A]) -> Bool
related(R:BinaryRelation[A], x:A, y:A) -> Bool
reflexive(R,A) := forall x in A: R(x,x)
symmetric(R,A) := forall x,y in A: R(x,y) => R(y,x)
transitive(R,A) := forall x,y,z in A: (R(x,y) and R(y,z)) => R(x,z)
equivalence(R,A) := reflexive(R,A) and symmetric(R,A) and transitive(R,A)
class_of(R,x) := {y in A | R(x,y)}
quotient(A,R) := {class_of(R,x) | x in A}

4. CORE FLOW
INPUT: A:Set[A], R:BinaryRelation[A]
PRECONDITION: R subset_of (A × A)
VALIDATE: reflexive(R,A), symmetric(R,A), transitive(R,A)
IF all true: classify R as EquivalenceRelation[A]
FOR EACH x in A: construct class_of(R,x)
DEDUPLICATE identical classes to form quotient(A,R)
OUTPUT: verified relation properties, equivalence classes, quotient set, partition certificate
ON FAILURE: report counterexample witnesses to failed properties

5. ALGEBRAIC LAWS
R(x,x)                                      [reflexivity]
R(x,y) <=> R(y,x)                           [symmetry]
R(x,y) and R(y,z) => R(x,z)                 [transitivity]
R(x,y) <=> class_of(R,x) = class_of(R,y)
class_of(R,x) = class_of(R,y) OR class_of(R,x) intersect class_of(R,y) = empty
union(quotient(A,R)) = A
forall C in quotient(A,R): C != empty
forall distinct C,D in quotient(A,R): C intersect D = empty

6. PARTITION CORRESPONDENCE
FROM_EQUIVALENCE: equivalence(R,A) => quotient(A,R) is a partition of A
FROM_PARTITION: for partition P of A, define R_P(x,y) iff exists C in P: x in C and y in C
THEN: equivalence(R_P,A)
BIJECTION: equivalence relations on A correspond one-to-one with partitions of A

7. INFERENCE RULES
REFLEXIVE: x in A |- R(x,x)
SYMMETRIC: R(x,y) |- R(y,x)
TRANSITIVE: R(x,y), R(y,z) |- R(x,z)
CLASS_MEMBERSHIP: R(x,y) |- y in class_of(R,x)
CLASS_IDENTITY: R(x,y) |- class_of(R,x) = class_of(R,y)
CLASS_DISJOINTNESS: not R(x,y) |- class_of(R,x) intersect class_of(R,y) = empty

8. EXAMPLES
INTEGER_CONGRUENCE: A=Z; R_n(a,b) iff n divides (a-b), n positive integer
QUOTIENT: Z/R_n has n residue classes
STRING_EQUIVALENCE: strings equivalent if they have the same length
CAUTION: similarity, proximity, or a merely symmetric relation need not be transitive

9. SEMANTIC RELATIONSHIPS
EquivalenceRelation IS_A BinaryRelation
EquivalenceRelation REQUIRES Reflexivity
EquivalenceRelation REQUIRES Symmetry
EquivalenceRelation REQUIRES Transitivity
EquivalenceRelation INDUCES EquivalenceClass
EquivalenceClass MEMBER_OF QuotientSet
QuotientSet CORRESPONDS_TO Partition
Representative REPRESENTS EquivalenceClass

10. SEMANTIC_EQUIVALENCE
SYNONYMS: equivalence relation; equivalence relation on a set
NOT_EQUIVALENT_TO: arbitrary binary relation; similarity relation; partial order
NORMALISATION: R(x,y) <=> x ~_R y when R is explicitly typed and scoped to A
CONTEXT: equivalence is relative to a specified relation, not necessarily object identity

11. VALIDATION
CHECK_DOMAIN: all related ordered pairs belong to A × A
CHECK_REFLEXIVITY: for each x in A, require (x,x) in R
CHECK_SYMMETRY: for each (x,y) in R, require (y,x) in R
CHECK_TRANSITIVITY: for each (x,y),(y,z) in R, require (x,z) in R
CHECK_PARTITION: every class nonempty; classes disjoint or equal; union equals A
FINITE_SETS: enumerate witnesses; complexity depends on representation
INFINITE_SETS: require proof or trusted symbolic reasoning, not finite sampling alone
EDGE_CASE: empty set has one binary relation (empty), which is an equivalence relation; its quotient is empty

12. CONFIDENCE
MATHEMATICAL_DEFINITIONS: 1.00 when verified under stated axioms
DERIVED_CLAIMS: 1.00 only with valid proof or exhaustive finite verification
HEURISTIC_CLASSIFICATION: record measured confidence separately; never treat as proof
UNKNOWN: explicitly label unverified claims

13. FLOW_HASH
HASH_INPUT: canonicalise(title, version, domain, signatures, axioms, rules, source_url)
HASH_ALGORITHM: SHA-256
FLOW_HASH: COMPUTE_AT_PUBLICATION (no fabricated digest)

14. UPDATE_MODEL
TRIGGER: definition change, corrected theorem, new examples, dependency revision
PROCESS: increment version; rerun property checks; regenerate canonical hash; retain provenance
COMPATIBILITY: preserve mathematical meanings of reflexivity, symmetry, transitivity

15. AI_CONSUMPTION
TASKS: relation typing; proof assistance; equivalence-class computation; quotient construction; partition validation; semantic retrieval
INPUT_REQUIREMENTS: explicit base set, relation predicate or ordered-pair set, assumptions
OUTPUT_REQUIREMENTS: typed result, proof or counterexample, assumptions, provenance, confidence
SAFETY: distinguish proved equivalence from approximate semantic similarity

16. PROVENANCE
This file is an original formalisation based on standard discrete mathematics.
The live source page was not inspected during creation; verify alignment with its editorial content before publication.