Semigroup Ontology

FLOW ALGEBRA
TITLE: Semigroup Ontology
DOMAIN: Discrete Mathematics > Algebraic Structures
SOURCE_URL: https://kirethresearch.org/booksai-core/discrete-mathematics-ontologies/algebraic-structures/semigroup-ontology.html
VERSION: 1.0
LANGUAGE: en-GB
AUTHORSHIP: Mark Higgins — Sky (Recursive AI)
SOURCE_STATUS: Formal mathematical synthesis; live page not independently verified.

1. PURPOSE
Represent semigroups as typed algebraic objects with machine-checkable associativity, substructures, homomorphisms, transformations and inference rules.

2. TYPED ENTITIES
Semigroup := (S: NonEmptySet, op: S × S -> S)
Element := member(S)
BinaryOperation := total closed map S × S -> S
Subsemigroup := nonempty T subseteq S, closed under op
Monoid := Semigroup with a two-sided identity
Group := Monoid with a two-sided inverse for every element
Homomorphism := map f: S -> T preserving operations
Congruence := equivalence relation compatible with op
Ideal := nonempty subset absorbing multiplication on the specified side
Idempotent := e in S with e*e=e
ZeroElement := z in S with z*x=x*z=z for all x in S

3. CORE AXIOMS
NONEMPTY: S != empty
CLOSURE: forall a,b in S: op(a,b) in S
ASSOCIATIVITY: forall a,b,c in S: op(op(a,b),c)=op(a,op(b,c))
No identity, inverse or commutativity is assumed.

4. OPERATIONS
MULTIPLY(a,b) := op(a,b)
POWER(a,n) := a repeated n times, n >= 1
GENERATED_SUBSEMIGROUP(A) := all nonempty finite products of elements of A
DIRECT_PRODUCT(S,T) := (S × T, componentwise multiplication)
OPPOSITE(S) := (S, op_rev) where op_rev(a,b)=op(b,a)
ADJOIN_IDENTITY(S) := S^1, with a new identity if S lacks one
QUOTIENT(S,~) := equivalence classes with induced operation, if ~ is a congruence

5. ALGEBRAIC LAWS
ASSOCIATIVE_REBRACKETING: (a*b)*c = a*(b*c)
POWER_ADDITION: a^m*a^n = a^(m+n), m,n >= 1
POWER_MULTIPLICATION: (a^m)^n = a^(mn), m,n >= 1
HOMOMORPHISM_PRESERVATION: f(a*b)=f(a) ◦ f(b)
HOMOMORPHISM_COMPOSITION: composition of semigroup homomorphisms is a homomorphism
IDEMPOTENT: e*e=e (only when e is marked idempotent)

6. STRUCTURAL RELATIONSHIPS
Semigroup IS_A AssociativeAlgebraicStructure
Monoid SPECIALISES Semigroup
Group SPECIALISES Monoid
Subsemigroup SUBSTRUCTURE_OF Semigroup
Homomorphism MAPS_BETWEEN Semigroup
Congruence ENABLES QuotientSemigroup
LeftIdeal / RightIdeal / TwoSidedIdeal SUBSETS_OF Semigroup

7. INFERENCE RULES
IF associative(op) AND closed(op,S) AND nonempty(S) THEN Semigroup(S,op).
IF Semigroup(S,op) AND exists e in S such that forall x, e*x=x*e=x THEN Monoid(S,op,e).
IF Semigroup(S,op) AND forall a,b: a*b=b*a THEN CommutativeSemigroup(S,op).
IF T nonempty subseteq S AND forall a,b in T: a*b in T THEN Subsemigroup(T,op).
IF f preserves multiplication AND is bijective THEN SemigroupIsomorphism(f).
IF ~ is an equivalence AND a~a' AND b~b' implies a*b~a'*b' THEN SemigroupCongruence(~).

8. EXAMPLES AND COUNTEREXAMPLES
EXAMPLE: Positive integers under addition form a commutative semigroup (without identity in the positive integers).
EXAMPLE: Nonempty strings under concatenation form a free semigroup.
EXAMPLE: Functions X->X under composition form a monoid and therefore a semigroup.
COUNTEREXAMPLE: Integers under subtraction do not form a semigroup because subtraction is not associative.
COUNTEREXAMPLE: A closed binary operation alone does not establish a semigroup without associativity.

9. VALIDATION
CHECK_NONEMPTY(S)
CHECK_TOTALITY_AND_CLOSURE(op,S)
CHECK_ASSOCIATIVITY(op,S)
For finite S: exhaustively test all triples (a,b,c) in S^3.
For infinite S: require symbolic proof or a trusted theorem, not finite sampling alone.
CHECK_SUBSEMIGROUP(T): nonempty and closed under op.
CHECK_HOMOMORPHISM(f): all relevant pairs preserve operation.
CHECK_QUOTIENT(~): equivalence and operation compatibility.
Report counterexample witnesses when a property fails.

10. SEMANTIC_EQUIVALENCE
'Semigroup' == 'Nonempty set with an associative binary operation'.
'Associative magma' == 'Semigroup' under the nonempty-set convention.
'Commutative semigroup' == 'Abelian semigroup'.
Do not conflate semigroup with monoid: a semigroup need not have an identity.

11. CONFIDENCE
Mathematical definitions and axioms: 1.00 (formal, conditional on stated conventions).
Specific finite-model claims: 1.00 only after exhaustive verification.
Unverified external source-specific claims: not asserted.

12. FLOW_HASH
MODEL: SHA-256 of canonical UTF-8 normalisation of the complete flow, excluding any stored hash value.
STATUS: Not computed; recompute after changes.

13. UPDATE_MODEL
TRIGGER: definition, axiom, relation, proof or source-page revision.
PROCESS: revise -> type-check -> validate -> recalculate FLOW_HASH -> version -> publish.
PROVENANCE: retain source URL, revision date, and change history.

14. AI_CONSUMPTION
INPUT: typed set S, binary operation op, optional subset/map/relation.
OUTPUT: semigroup classification, derived properties, proofs or counterexamples.
REASONING: apply axioms and inference rules; distinguish established facts from assumptions.
SAFETY: never infer commutativity, identity or inverses from associativity alone.
INTEROPERABILITY: link to Monoid, Group, Homomorphism, Congruence and Algebraic Structure ontologies.