N-ary Relation Ontology

FLOW ALGEBRA
N-ARY RELATION ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Relation Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/relation-entities/n-ary-relation-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Status: Formal ontology constructed from established mathematics; source page not independently verified.
Version: 1.0 | Date: 2026-10-13

01. PURPOSE
Represent typed n-ary relations, ordered tuples, interpretation, transformations, constraints and inference without losing argument roles or order.

02. SIGNATURE
sort Entity; sort Domain; sort Relation; sort Tuple; sort Proposition.
NAT n >= 0.
R : Relation(D1,...,Dn) where Di are nonempty carrier sets unless explicitly declared otherwise.
Tuple t=(x1,...,xn) with xi in Di.
R subseteq D1 x ... x Dn.
member(t,R) : Boolean.
arity(R)=n.

03. SEMANTIC DEFINITION
R(x1,...,xn) iff (x1,...,xn) in R.
Arguments are positional; permuting arguments changes meaning unless a symmetry invariant is established.
A relation is a set of tuples, not merely a text label or untyped edge.
For n=2 it is binary; for n=1 unary; for n=0 it is a truth-valued nullary relation.

04. ENTITY SCHEMA
relation_id: stable identifier
label: human-readable name
arity: nonnegative integer
argument_roles: ordered array of length arity
argument_domains: ordered array of length arity
extension: set of well-typed tuples, or symbolic predicate
intension: optional defining rule
context: interpretation, units and assumptions
provenance: source and version
confidence: evidence quality, not mathematical truth

05. VALIDITY
valid(R) iff arity(R)=length(roles)=length(domains).
valid_tuple(t,R) iff length(t)=arity(R) and forall i, t[i] in Di.
member(t,R) requires valid_tuple(t,R).
Duplicate tuples have no multiplicity in set semantics.
An empty extension is permitted.

06. EXAMPLES
Between(x,y,z): x lies between y and z in a specified ordered or geometric context.
SumEquals(x,y,z): x+y=z over a stated numeric domain.
Enrolled(student,course,term): ternary institutional relation.
Purchase(buyer,item,seller,time): quaternary event relation.
These are examples, not universally applicable axioms.

07. PROJECTION
For I=(i1,...,ik) distinct indices from 1..n:
project_I(R)={ (t[i1],...,t[ik]) | t in R }.
Projection may discard correlations and cannot generally reconstruct R.

08. SELECTION
select_P(R)={t in R | P(t)} where P is a well-typed predicate.
select_true(R)=R; select_false(R)=empty_set.

09. PERMUTATION
For permutation pi of 1..n:
permute_pi(R)={ (t[pi(1)],...,t[pi(n)]) | t in R }.
The output domain signature is permuted accordingly.
permute_inverse(pi)(permute_pi(R))=R.

10. PRODUCT AND JOIN
R x S contains concatenated tuples from R and S.
Natural join requires explicit shared-attribute identity and compatible domains.
Theta join filters a Cartesian product by a declared predicate.
Do not infer a join merely from matching display labels.

11. UNION, INTERSECTION, DIFFERENCE
For relations R,S with the same typed signature:
R union S={t | t in R or t in S}.
R intersect S={t | t in R and t in S}.
R minus S={t | t in R and t not in S}.
Incompatible signatures require an explicit coercion or mapping.

12. COMPLEMENT
Relative complement of R is (D1 x ... x Dn) minus R.
Complement is context-dependent on declared carrier sets.
Do not assume a global universe of all possible tuples.

13. EXISTENTIAL QUANTIFICATION
exists xi R(x1,...,xn) corresponds to projection away from argument i.
Quantification must preserve all remaining argument roles.

14. UNIVERSAL QUANTIFICATION
forall xi in Di, R(x1,...,xn) means every admissible value at position i satisfies R.
For empty Di, universal claims are vacuously true under classical semantics.

15. FUNCTIONAL DEPENDENCY
For position sets X,Y: X -> Y holds iff tuples agreeing on X agree on Y.
Functional dependency is a property to validate, not a default of n-ary relations.

16. REIFICATION
A relation instance may be represented as an event/entity e with role edges.
Reification preserves tuple identity and supports time, provenance and qualifiers.
Reconstruction requires a unique tuple key or explicit duplicate-handling policy.

17. GRAPH REPRESENTATION
Node RelationDefinition: signature, arity, role ordering.
Node RelationInstance: instance identifier.
Edges: HAS_ARGUMENT_1 through HAS_ARGUMENT_n, INSTANCE_OF, HAS_CONTEXT.
A simple binary-edge graph cannot faithfully encode all n-ary roles without reification or hyperedges.

18. SEMANTIC EQUIVALENCE
R equivalent_to S only under an explicit role-preserving bijection and equal extensions in a shared interpretation.
Equal labels do not imply equal relations.
Renaming bound variables preserves meaning if capture is avoided.
Argument permutations require proof of invariance or a declared role mapping.

19. INFERENCE RULES
R(t) and rule forall x (R(x) -> Q(x)) imply Q(t), subject to type correctness.
R(t) implies project_I(R)(project_I(t)).
R(t) and P(t) imply select_P(R)(t).
R(t) does not imply R(permute(t)) without a symmetry axiom.
Open-world absence of a tuple is not proof of its negation.

20. CONSTRAINTS
Check arity, domains, role uniqueness when required, admissible values and declared keys.
Check join compatibility, provenance and temporal consistency where relevant.
For numerical predicates, check units and exact versus approximate comparison semantics.

21. CONFIDENCE
Mathematical theorems: confidence tied to formal proof status.
Observed relation instances: confidence tied to source evidence.
Never treat probabilistic confidence as a substitute for logical validity.

22. FLOW HASH
Canonicalize ontology version, role order, domains, rule text and extension representation.
FLOW_HASH = SHA256(canonical_UTF8_payload).
Hash computation is an implementation instruction; no precomputed digest is asserted here.

23. UPDATE MODEL
Track revision_id, timestamp, changed signatures, added/removed tuples, changed rules and provenance.
Breaking changes include arity or role-order changes.
Revalidate dependent queries, joins and inference rules after a breaking change.

24. AI CONSUMPTION
Input: typed relation definition, tuples, context, requested operation.
Process: validate -> normalize -> apply relation algebra -> infer -> verify.
Output: typed results, rule trace, assumptions, provenance, confidence and errors.
Never fabricate relation membership from an ontology label alone.

25. TEST CASES
R={(1,2,3),(1,4,5)} over integers^3.
arity(R)=3.
project_(1,2)(R)={(1,2),(1,4)}.
select_(third>3)(R)={(1,4,5)}.
permute_(2,1,3)(R)={(2,1,3),(4,1,5)}.
Invalid: R(1,2) due to arity mismatch.

26. ONTOLOGY LINKS
BROADER: Mathematical Relation Entity Ontology; Relation Entities.
RELATED: Binary Relation; Cartesian Product; Set; Tuple; Predicate; Function; Database Relation; Hypergraph; Knowledge Graph; Constraint.
DISTINGUISH: An n-ary relation need not be functional, symmetric, total or nonempty.

27. TERMINATION
Emit validation status, typed relation result and provenance record.
END FLOW ALGEBRA