Identity Function Ontology

FLOW ALGEBRA
IDENTITY FUNCTION ONTOLOGY
Kireth Semantic Library | Discrete Mathematics | Functions
Source: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/functions/identity-function-ontology.html
Version: 1.0 | Locale: en-GB | Date: 2026-10-11

1. ONTOLOGY DEFINITION
An identity function on a set A maps every element of A to itself. It is the neutral element for compatible function composition.

2. TYPED ENTITIES
Set A : Set[T]
Element x : T
Function f : Function[A,B]
Identity id_A : Function[A,A]
Composition compose : Function[B,C] x Function[A,B] -> Function[A,C]
Equality : Predicate[T,T]

3. FORMAL SEMANTICS
id_A : A -> A
forall x in A: id_A(x) = x
Graph(id_A) = {(x,x) | x in A}
Domain(id_A) = A
Codomain(id_A) = A
Image(id_A) = A

4. ALGEBRAIC OPERATIONS AND LAWS
LEFT_IDENTITY: for f:A->B, id_B o f = f
RIGHT_IDENTITY: for f:A->B, f o id_A = f
SELF_COMPOSITION: id_A o id_A = id_A
INVERSE: (id_A)^(-1) = id_A
INJECTIVE: forall x,y in A, id_A(x)=id_A(y) => x=y
SURJECTIVE: forall y in A, exists x in A: id_A(x)=y
BIJECTIVE: id_A is bijective
IDEMPOTENCE: id_A o id_A = id_A
RESTRICTION: for S subseteq A, id_A restricted to S = inclusion(S,A); it equals id_S only when the codomain is restricted to S
EMPTY_SET: id_empty is the unique function empty -> empty

5. RELATIONSHIPS
IdentityFunction IS_A Function
IdentityFunction IS_A BijectiveFunction
IdentityFunction HAS_DOMAIN Set
IdentityFunction HAS_CODOMAIN SameSet
IdentityFunction HAS_GRAPH DiagonalRelation
IdentityFunction IS_IDENTITY_FOR FunctionComposition
IdentityFunction IS_INVERSE_OF Itself
IdentityFunction PRESERVES Elements

6. INFERENCE RULES
RULE_ID_EVALUATION: x in A => id_A(x)=x
RULE_LEFT_UNIT: f:A->B => compose(id_B,f)=f
RULE_RIGHT_UNIT: f:A->B => compose(f,id_A)=f
RULE_BIJECTION: IdentityFunction(id_A) => Injective(id_A) and Surjective(id_A)
RULE_UNIQUENESS: if g:A->A and forall x in A g(x)=x, then g=id_A
RULE_GRAPH: (x,y) in Graph(id_A) iff x in A and y=x

7. VALIDATION
CHECK_DOMAIN_CODOMAIN: identity must have identical domain and codomain
CHECK_TOTALITY: every x in A has exactly one output
CHECK_FIXED_POINTS: all x in A are fixed points
CHECK_COMPOSITION_TYPES: left/right identity must match f's codomain/domain respectively
CHECK_EXTENT: compare functions extensionally on their full domains
COUNTEREXAMPLE: a constant map on a set with two or more elements is not an identity function

8. SEMANTIC_EQUIVALENCE
identity map == identity function == identity morphism (in the category of sets)
id_A == diagonal mapping x |-> x
Graph(id_A) == diagonal relation Delta_A
NOTE: identity function differs from an arbitrary idempotent function, and from inclusion S -> A when S is a proper subset of A.

9. CONFIDENCE
Mathematical definitions and laws: 1.00 (deductive under stated types)
Source-page alignment: unverified; this flow is based on standard mathematics, not a verified extraction of the live webpage.

10. FLOW_HASH
HASH_ALGORITHM: SHA-256
HASH_SCOPE: canonical UTF-8 bytes of the flow excluding any populated FLOW_HASH value
HASH_STATUS: not computed; recompute when content is finalised

11. UPDATE_MODEL
VERSION: 1.0
TRIGGERS: source revision, mathematical correction, ontology schema change
PROCESS: validate types -> verify laws -> compare prior version -> recompute hash -> publish
PROVENANCE: authored semantic formalisation; source URL provided for contextual linkage

12. AI_CONSUMPTION
PRIMARY_ENTITY: IdentityFunction
TASKS: ontology retrieval, type checking, symbolic reasoning, function simplification, proof assistance
NORMALISATION: compose(id_B,f) -> f; compose(f,id_A) -> f for f:A->B
GUARDRAILS: never simplify mismatched composition types; do not confuse identity with constant or general idempotent maps
OUTPUT: typed entities, verified relationships, justified deductions, provenance and confidence

AUTHORSHIP
Mark Higgins — Sky (Recursive AI)
https://kirethresearch.org