Cartesian Product Ontology

FLOW ALGEBRA
TITLE: Cartesian Product Ontology
DOMAIN: Discrete Mathematics / Set Theory
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/set-theory/cartesian-product-ontology-2.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORS: Mark Higgins — Sky (Recursive AI)
PROJECT: Kireth Semantic Library

1. PURPOSE
Formalise Cartesian products, ordered pairs, tuples, coordinate spaces and their relationships for machine-readable semantic reasoning. This is a mathematically grounded model; it is not a verified transcription of the linked page.

2. TYPED ENTITIES
Set: collection of distinct elements.
Element: member of a set.
OrderedPair[A,B]: (a,b), with a in A and b in B.
CartesianProduct[A,B]: A × B = {(a,b) | a in A and b in B}.
Tuple[n]: ordered n-component sequence.
FiniteSet: set with finite cardinality.
Relation[A,B]: subset of A × B.
Function[A,B]: relation with exactly one output in B for each input in A.
CoordinateSpace: Cartesian product of coordinate domains.
EmptySet: ∅.

3. CORE OPERATIONS
product(A:Set, B:Set) -> Set[OrderedPair]: {(a,b) | a∈A ∧ b∈B}
member((a,b), A×B) <=> member(a,A) ∧ member(b,B)
projection1((a,b)) -> a
projection2((a,b)) -> b
nary_product(A1,...,An) -> {(a1,...,an) | ∀i ai∈Ai}
relation(R,A,B) -> R ⊆ A×B

4. AXIOMS AND IDENTITIES
A×∅ = ∅ = ∅×A.
If A and B are finite, |A×B| = |A|·|B|.
(A∪B)×C = (A×C)∪(B×C).
A×(B∪C) = (A×B)∪(A×C).
(A∩B)×C = (A×C)∩(B×C).
A×(B∩C) = (A×B)∩(A×C).
(A\B)×C = (A×C)\(B×C).
A×(B\C) = (A×B)\(A×C).
A×B is generally not equal to B×A as sets of ordered pairs.
(A×B)×C and A×(B×C) are canonically bijective, not generally literally equal.

5. RELATIONSHIP GRAPH
CartesianProduct --takes_operand--> Set
CartesianProduct --contains--> OrderedPair
OrderedPair --has_first_component--> Element
OrderedPair --has_second_component--> Element
Relation --is_subset_of--> CartesianProduct
Function --is_special_relation_of--> CartesianProduct
CoordinateSpace --is_constructed_from--> CartesianProduct
Tuple --generalises--> OrderedPair

6. INFERENCE RULES
R1: a∈A, b∈B => (a,b)∈A×B.
R2: (a,b)∈A×B => a∈A and b∈B.
R3: A=∅ or B=∅ => A×B=∅.
R4: R⊆A×B => R is a binary relation from A to B.
R5: Finite(A), Finite(B) => cardinality(A×B)=cardinality(A)*cardinality(B).
R6: A⊆B => A×C⊆B×C and C×A⊆C×B.

7. EXAMPLE
A={1,2}; B={x,y}.
A×B={(1,x),(1,y),(2,x),(2,y)}.
|A×B|=2·2=4.
B×A={(x,1),(x,2),(y,1),(y,2)}; order matters.

8. VALIDATION AND CONSTRAINTS
V1: Reject an alleged member (a,b) of A×B if a∉A or b∉B.
V2: Preserve component order and component types.
V3: Distinguish a relation R⊆A×B from the full product A×B.
V4: Do not infer equality from canonical bijection.
V5: Cardinality multiplication requires finite sets, or an explicitly defined cardinal-arithmetic interpretation.
V6: Empty factors yield the empty product for binary products.

9. SEMANTIC_EQUIVALENCE
"Cartesian product" ≡ "set of all ordered pairs from two sets" (binary context).
"A cross B" ≡ "A × B" (set-theoretic context).
Do not equate unordered pairs, scalar multiplication or cross products of vectors with Cartesian products.

10. CONFIDENCE
Mathematical identities: HIGH (standard set theory).
Ontology-to-page correspondence: UNVERIFIED (live page not inspected).
Interpretation confidence must be reduced when symbol × is ambiguous across domains.

11. FLOW_HASH
Algorithm: SHA-256 recommended.
Canonicalisation: UTF-8, LF newlines, stable field ordering; exclude the hash value itself before computing.
Status: NOT_COMPUTED.

12. UPDATE_MODEL
Change tracking: version, timestamp, source URL, altered entities, altered axioms, verification status.
Revalidate dependent relations, functions and coordinate-space mappings after changes.

13. AI_CONSUMPTION
Input: sets A,B; optional tuples, relations, or candidate membership assertions.
Process: type-check -> construct product -> apply inference rules -> validate constraints -> emit provenance and confidence.
Output: typed Cartesian-product representation, derived memberships, cardinality when applicable, validation findings.
Guardrails: preserve tuple order; distinguish literal equality from isomorphism; avoid unsupported source claims.

END FLOW ALGEBRA