Symmetric Relation Ontology

FLOW ALGEBRA
SYMMETRIC RELATION ONTOLOGY
Kireth Semantic Library | Mathematical Entities > Relation Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/relation-entities/symmetric-relation-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Date: 2026-10-13
Status: Formal mathematical synthesis; source page not independently verified.

01. ONTOLOGY IDENTITY
ontology_id: kireth.math.relation.symmetric.v1
entity_type: SymmetricRelation
parent_type: BinaryRelation
subject_domain: SetTheory / RelationAlgebra / DiscreteMathematics

02. CORE DEFINITION
Let A be a set and R subseteq A x A.
symmetric(R,A) iff forall x,y in A: ((x,y) in R => (y,x) in R).
Symmetry does not imply reflexivity or transitivity.
The empty relation is symmetric on every set.

03. TYPED ENTITIES
Set: A
Element: x,y in A
OrderedPair: (x,y)
BinaryRelation: R subseteq A x A
ConverseRelation: R_inverse = {(y,x) | (x,y) in R}
SymmetricClosure: R_sym = R union R_inverse

04. RELATIONSHIP GRAPH
SymmetricRelation --is_a--> BinaryRelation
SymmetricRelation --has_domain--> Set
SymmetricRelation --has_property--> Symmetry
SymmetricRelation --equals_its_converse--> ConverseRelation
SymmetricClosure --extends--> BinaryRelation
EquivalenceRelation --requires--> SymmetricRelation

05. FORMAL AXIOM
forall x in A:
  forall y in A:
    R(x,y) -> R(y,x)

06. CONVERSE CHARACTERISATION
symmetric(R,A) <=> R = R_inverse.
Equivalently: R_inverse subseteq R, which also implies the reverse inclusion by taking converses.

07. INFERENCE RULES
rule symmetry_forward:
  premises: R(x,y), symmetric(R,A), x in A, y in A
  conclude: R(y,x)
rule symmetry_test:
  premises: forall (x,y) in R, (y,x) in R
  conclude: symmetric(R,A)
rule asymmetry_counterexample:
  premises: R(x,y), not R(y,x)
  conclude: not symmetric(R,A)
  condition: closed_world_relation_known_complete

08. SYMMETRIC CLOSURE
closure_sym(R) = R union R_inverse.
It is the unique least symmetric relation containing R.
closure_sym(closure_sym(R)) = closure_sym(R).
If R subseteq S then closure_sym(R) subseteq closure_sym(S).

09. ALGEBRAIC PROPERTIES
If R and S are symmetric on A, then R union S and R intersection S are symmetric.
The complement (A x A) \ R is symmetric when R is symmetric.
Composition of symmetric relations need not be symmetric.
(R o S)_inverse = S_inverse o R_inverse.
If R,S are symmetric and commute under composition, R o S is symmetric.

10. GRAPH INTERPRETATION
A symmetric relation corresponds to a directed graph in which every directed edge has its reverse.
Loops are permitted but not required.
For a finite relation, adjacency matrix M satisfies M = transpose(M).

11. COMPARISON WITH RELATED PROPERTIES
Reflexive: forall x in A, R(x,x).
Irreflexive: forall x in A, not R(x,x).
Antisymmetric: R(x,y) and R(y,x) imply x=y.
Asymmetric: R(x,y) implies not R(y,x).
Transitive: R(x,y) and R(y,z) imply R(x,z).
An equivalence relation is reflexive, symmetric, and transitive.
A relation may be symmetric and irreflexive (e.g. distinct-person friendship).
A relation symmetric and asymmetric must be empty.
A relation symmetric and antisymmetric contains only diagonal pairs.

12. EXAMPLES
A={1,2,3}; R={(1,2),(2,1),(3,3)} is symmetric.
A={1,2}; R={(1,2)} is not symmetric.
Equality on A is symmetric, reflexive, and transitive.
The relation x != y is symmetric and irreflexive, but generally not transitive.

13. FINITE VALIDATION ALGORITHM
function is_symmetric(A,R):
  require R subseteq A x A
  for each (x,y) in R:
    if (y,x) not in R: return false
  return true
Time: O(|R|) expected with hashed membership; memory depends on relation representation.

14. SEMANTIC EQUIVALENCE
'Symmetric binary relation' == 'Relation equal to its converse' (under a fixed domain and conventional converse).
'Undirected adjacency' is a graph interpretation, not an unconditional synonym for all relation semantics.
Preserve the distinction between symmetric and antisymmetric.

15. KNOWLEDGE REPRESENTATION
triple: (R, rdf:type, SymmetricRelation)
triple: (R, hasDomain, A)
triple: (R, hasProperty, Symmetry)
constraint: every asserted R(x,y) requires R(y,x) in the completed extension.
Open-world systems must not infer non-symmetry merely from an unasserted reverse pair.

16. CONFIDENCE
formal_definition: 1.00
mathematical_inference_rules: 1.00 under stated premises
source_page_alignment: unverified
empirical_graph_claims: require provenance

17. VALIDATION
validate_domain: R subseteq A x A
validate_pair_reversal: forall (x,y) in R, (y,x) in R
validate_closure: R subseteq closure_sym(R)
validate_idempotence: closure_sym(closure_sym(R)) = closure_sym(R)
validate_converse: symmetric(R,A) iff R = R_inverse

18. FLOW HASH
hash_algorithm: SHA-256
hash_scope: canonicalised ontology body, excluding stored hash field
canonicalisation: UTF-8, LF newlines, normalised whitespace and deterministic field ordering
FLOW_HASH: compute_at_publication

19. UPDATE MODEL
version: 1.0
change_policy: semantic versioning
update_trigger: corrected theorem, expanded example, new relationship or formal schema revision
preserve_provenance: true

20. AI CONSUMPTION
input: relation R and domain A, with completeness policy
outputs: symmetry classification, counterexamples, symmetric closure, inferred reversed pairs
reasoning_mode: deterministic symbolic reasoning
safety: distinguish absence of evidence from evidence of absence
trace: retain source pairs and rule identifiers for every inferred pair

21. QUALITY ASSURANCE TESTS
test_1: empty relation => symmetric
test_2: singleton loop => symmetric
test_3: one-way edge => not symmetric (complete relation)
test_4: two-way edge => symmetric
test_5: symmetric closure contains original pairs and their converses
test_6: no inferred reflexivity or transitivity from symmetry alone

END FLOW ALGEBRA