Semigroup Ontology

FLOW ALGEBRA
SEMIGROUP ONTOLOGY — KIRETH SEMANTIC LIBRARY
Version: 4.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/algebraic-entities/semigroup-ontology-2.html
Status: Independently constructed mathematical model; source-page alignment not verified.
Authorship: Mark Higgins — Sky (Recursive AI)
Website: https://kirethresearch.org

01. ONTOLOGY IDENTITY
ONTOLOGY_ID: mathematical.algebraic.semigroup
ENTITY_TYPE: AlgebraicStructure
PARENT: AlgebraicStructure
RELATED: Magma, Monoid, Group, Subsemigroup, Ideal, Homomorphism
DEFINITION: A semigroup is a nonempty set S equipped with an associative binary operation *: S × S -> S.

02. TYPED ENTITIES
Semigroup := (carrier:Set[Element], operation:BinaryOperation, associative:Boolean)
Element := member(carrier)
BinaryOperation := function(carrier × carrier -> carrier)
Subsemigroup := subset(carrier) closed under operation and nonempty
Idempotent := element e satisfying e*e=e
Identity := element e satisfying forall x: e*x=x*e=x
Zero := element z satisfying forall x: z*x=x*z=z
Homomorphism := map preserving operation
Ideal := subset absorbing multiplication on specified side
Congruence := equivalence relation compatible with multiplication

03. CORE AXIOMS
AXIOM S1 NONEMPTY: S != empty_set.
AXIOM S2 CLOSURE: forall a,b in S, a*b in S.
AXIOM S3 ASSOCIATIVITY: forall a,b,c in S, (a*b)*c = a*(b*c).
NOTE: Identity, inverses, cancellation and commutativity are not required.
NOTE: Associativity makes an unparenthesised finite product well-defined for a fixed ordered list.

04. SEMANTIC RELATIONSHIPS
Semigroup IS_A AlgebraicStructure
Semigroup HAS_CARRIER Set
Semigroup HAS_OPERATION BinaryOperation
BinaryOperation SATISFIES Associativity
Monoid IS_A Semigroup
Group IS_A Monoid
CommutativeSemigroup IS_A Semigroup
Band IS_A IdempotentSemigroup
Subsemigroup SUBSTRUCTURE_OF Semigroup
SemigroupHomomorphism PRESERVES BinaryOperation
Congruence COMPATIBLE_WITH BinaryOperation
QuotientSemigroup DERIVED_FROM Semigroup AND Congruence

05. SUBTYPES AND CONSTRUCTIONS
COMMUTATIVE: forall a,b, a*b=b*a.
IDEMPOTENT (BAND): forall a, a*a=a.
LEFT_ZERO: forall a,b, a*b=a.
RIGHT_ZERO: forall a,b, a*b=b.
NULL_SEMIGROUP: all products equal a fixed zero element.
MONOID: semigroup with two-sided identity.
GROUP: monoid with two-sided inverses for all elements.
FREE_SEMIGROUP(A): nonempty finite words over nonempty alphabet A under concatenation.
PRODUCT: Cartesian product of semigroups with componentwise operation.
OPPOSITE: a *op b := b*a; associative whenever original is associative.

06. SUBSEMIGROUP AND IDEALS
SUBSEMIGROUP_TEST: T subset S, T nonempty, forall a,b in T: a*b in T.
LEFT_IDEAL: I nonempty subset S and S*I subset I.
RIGHT_IDEAL: I nonempty subset S and I*S subset I.
TWO_SIDED_IDEAL: left_ideal(I) AND right_ideal(I).
GENERATED_SUBSEMIGROUP(X): intersection of all subsemigroups containing X, for nonempty X.
PRINCIPAL_IDEALS: when S lacks identity use S^1, obtained by adjoining an identity if needed.
GREEN_RELATIONS: L,R,J,H,D defined through principal ideals and their intersections/compositions.

07. MORPHISMS
HOMOMORPHISM f:S->T: forall a,b in S, f(a*b)=f(a)·f(b).
ISOMORPHISM: bijective homomorphism.
ENDOMORPHISM: homomorphism S->S.
AUTOMORPHISM: bijective endomorphism.
KERNEL_CONGRUENCE: a ~ b iff f(a)=f(b).
QUOTIENT: equivalence classes under congruence inherit a well-defined associative operation.
FIRST_ISOMORPHISM: S/kernel(f) is isomorphic to image(f).

08. FORMAL INFERENCE RULES
R01: magma(S) AND associative(op) -> semigroup(S,op).
R02: semigroup(S,op) -> closed(op,S).
R03: semigroup(S,op) -> associative(op).
R04: semigroup(S,op) AND identity(e) -> monoid(S,op,e).
R05: monoid(S) AND every_element_invertible(S) -> group(S).
R06: semigroup(S) AND commutative(op) -> commutative_semigroup(S).
R07: semigroup(S) AND forall x x*x=x -> band(S).
R08: semigroup(S) AND nonempty(T) AND T subset S AND closed(op,T) -> subsemigroup(T,S).
R09: subsemigroup(T,S) -> semigroup(T,op_restricted).
R10: semigroup(S) AND homomorphism(f,S,T) -> image(f) subsemigroup_of T.
R11: homomorphism(f,S,T) -> kernel_relation(f) congruence_on S.
R12: congruence(~,S) -> quotient(S,~) semigroup.
R13: semigroup(S) AND semigroup(T) -> direct_product(S,T) semigroup.
R14: semigroup(S) -> opposite(S) semigroup.
R15: semigroup(S) AND zero(z) -> idempotent(z).
R16: semigroup(S) AND identity(e) -> idempotent(e).
R17: group(S) -> monoid(S).
R18: monoid(S) -> semigroup(S).
R19: finite_semigroup(S) -> exists idempotent in S.
R20: associative(op) AND nonempty(A) -> free_semigroup(A) under concatenation.
R21: homomorphism(f,S,T) AND homomorphism(g,T,U) -> homomorphism(g∘f,S,U).
R22: isomorphism(f,S,T) -> semigroup(S) iff semigroup(T).
R23: left_ideal(I,S) AND right_ideal(I,S) -> two_sided_ideal(I,S).
R24: semigroup(S) AND commutative(op) -> left_ideal(I,S) iff right_ideal(I,S).
R25: identity(e,S) AND identity(f,S) -> e=f.
R26: homomorphism(f,S,T) AND idempotent(e,S) -> idempotent(f(e),T).
R27: semigroup(S) AND finite(S) -> for each a, some powers a^m=a^n with m<n.
R28: semigroup(S) AND cancellation(S) AND finite(S) -> group(S).
R29: group(S) -> cancellation(S).
R30: semigroup(S) AND regular(a) -> exists x in S: a=a*x*a.

09. VALIDATION CONSTRAINTS
V01: carrier must be nonempty.
V02: operation must be total on carrier × carrier.
V03: output of operation must belong to carrier.
V04: associativity must hold for all triples.
V05: identity cannot be inferred from associativity alone.
V06: commutativity cannot be inferred from associativity alone.
V07: a subsemigroup must be nonempty and closed.
V08: a quotient operation requires a congruence, not merely an arbitrary equivalence relation.
V09: a homomorphism need not be injective or surjective.
V10: for infinite semigroups, existence of an idempotent is not guaranteed.
V11: cancellation alone does not imply group structure for infinite semigroups.
V12: an ideal need not be a subgroup or contain an identity.

10. MATHEMATICAL EXAMPLES
E01: (positive integers,+) is a commutative semigroup without an identity in its carrier.
E02: (nonnegative integers,+) is a commutative monoid.
E03: (integers,+) is an Abelian group and hence a semigroup.
E04: (nonempty strings,concatenation) is a free semigroup.
E05: (all strings including empty string,concatenation) is a monoid.
E06: (positive integers,multiplication) is a commutative semigroup.
E07: (functions X->X,composition) is a monoid when X is nonempty.
E08: (nonempty subsets of X,union) is a commutative idempotent semigroup when X is nonempty.
E09: (integers,max) is a commutative idempotent semigroup.
E10: (positive integers,subtraction) is not a semigroup: closure and associativity fail.
E11: (real 2×2 matrices,multiplication) is a generally noncommutative monoid.
E12: left-zero multiplication a*b=a on any nonempty set defines a semigroup.

11. KNOWLEDGE GRAPH TRIPLES
(Semigroup, rdf:type, AlgebraicStructure)
(Semigroup, requires, Associativity)
(Semigroup, requires, Closure)
(Monoid, subclassOf, Semigroup)
(Group, subclassOf, Monoid)
(Band, subclassOf, Semigroup)
(SemigroupHomomorphism, preserves, BinaryOperation)
(Subsemigroup, subsetOf, SemigroupCarrier)
(Congruence, enables, QuotientSemigroup)
(FreeSemigroup, operation, Concatenation)

12. SEMANTIC EQUIVALENCE
Semigroup ≡ associative magma (with nonempty carrier and total binary operation).
Associative binary system ≈ semigroup only if nonempty closure and totality are established.
Semigroup != monoid unless identity is present.
Semigroup != group unless identity and inverses are present.
Subsemigroup != subgroup unless group axioms are satisfied on subset.

13. CONFIDENCE
CONFIDENCE mathematical_axioms: 1.00
CONFIDENCE standard_examples: 0.99
CONFIDENCE inferred_properties: conditional_on_verified_premises
CONFIDENCE source_page_alignment: unverified

14. FLOW HASH
FLOW_HASH algorithm: SHA-256
FLOW_HASH scope: canonical UTF-8 representation of the final published flow
FLOW_HASH value: COMPUTE_ON_PUBLICATION
FLOW_HASH note: no fabricated checksum; regenerate after each content change.

15. UPDATE MODEL
UPDATE_MODEL version: 4.0
UPDATE_MODEL mode: versioned_append_and_validate
UPDATE_MODEL triggers: source_page_revision, theorem_correction, relation_update, example_update
UPDATE_MODEL requirements: retain provenance, run consistency checks, recompute flow hash

16. AI CONSUMPTION
AI_CONSUMPTION mode: typed_semantic_reasoning
AI_CONSUMPTION tasks: classification, theorem_prerequisite_checking, structure_validation, knowledge_graph_linking
AI_CONSUMPTION procedure:
  1. Parse carrier and operation with explicit types.
  2. Verify nonempty carrier, totality, closure, associativity.
  3. Infer optional properties only when separately proven.
  4. Apply inference rules with premise tracking.
  5. Preserve distinctions among semigroup, monoid and group.
  6. Flag counterexamples and unverified source-specific claims.
  7. Return provenance, confidence and validation status.

END FLOW ALGEBRA