Inequality Relationship Ontology

FLOW ALGEBRA
TITLE: Inequality Relationship Ontology
VERSION: 1.0
SOURCE: https://kirethresearch.org/booksai-core/ontologies/relationship-ontologies/mathematical-relationships/inequality-relationship-ontology.html
AUTHORS: Mark Anthony Higgins — Sky (Recursive AI)
STATUS: Companion formalisation; source page not independently verified.

flow "InequalityRelationshipOntology" v1.0

ENTITY_TYPES:
  Quantity: {id: string, value: number?, unit: string?, domain: string}
  Expression: {id: string, syntax: string, domain: string}
  Inequality: {id: string, left: -Expression, operator: Operator, right: -Expression, context: string}
  Evidence: {id: string, source: string, timestamp: string?}

OPERATOR:
  LT: "<"
  LE: "<="
  GT: ">"
  GE: ">="
  NE: "!="

RELATIONSHIPS:
  less_than(a,b)        := a < b
  less_or_equal(a,b)    := a <= b
  greater_than(a,b)     := a > b
  greater_or_equal(a,b) := a >= b
  not_equal(a,b)        := a != b

PRECONDITIONS:
  - Comparisons require compatible mathematical domains.
  - Ordered comparisons require a declared order relation.
  - Quantity comparisons require compatible units and conversion provenance.
  - Floating-point comparisons must declare tolerance and precision policy.

INFERENCE_RULES:
  converse:     a < b  => b > a
  converse_eq:  a <= b => b >= a
  transitive:   a < b AND b < c => a < c
  mixed:        a < b AND b <= c => a < c
  nonstrict:    a <= b AND b <= c => a <= c
  irreflexive:  NOT(a < a)
  antisymmetry: a <= b AND b <= a => a = b
  asymmetry:    a < b => NOT(b < a)
  addition:     a < b => a+c < b+c
  multiply_positive: a < b AND c > 0 => a*c < b*c
  multiply_negative: a < b AND c < 0 => a*c > b*c
  distinction:  a != b does NOT imply a < b without total ordering

SEMANTIC_EQUIVALENCE:
  - "less than" <=> "strictly below" within the same order context
  - "at most" <=> "less than or equal to"
  - "at least" <=> "greater than or equal to"
  - "not equal" is not synonymous with either "less than" or "greater than" in partial orders

CONFIDENCE:
  value: 0.0..1.0
  basis: [formal_proof, measured_observation, model_estimate, asserted_claim]
  rule: A formal derivation is valid only if all premises and domain assumptions hold.
  rule: Confidence scores are metadata, not replacements for truth values.

PROVENANCE:
  capture: [source, author, evidence_id, derivation, timestamp, domain_assumptions]

VALIDATION:
  - Reject undefined operators and incompatible units.
  - Reject strict self-inequality in an irreflexive order.
  - Detect inconsistent claims (a < b AND b <= a) in the same context.
  - Do not assume trichotomy for partial orders.
  - Preserve strict versus non-strict comparison in transformations.

FLOW_HASH:
  algorithm: SHA-256
  input: canonical_serialisation(ontology_version, entities, relations, rules, assumptions)
  output: deterministic_content_digest

UPDATE_MODEL:
  strategy: versioned_incremental
  on_change: [revalidate_affected_relations, recompute_inferences, update_provenance, regenerate_flow_hash]
  preserve: [prior_versions, audit_history]

AI_CONSUMPTION:
  format: typed_semantic_relations
  tasks: [mathematical_reasoning, constraint_solving, knowledge_graph_validation, comparison, explanation]
  safeguards: [check_domain, check_units, check_assumptions, cite_provenance, distinguish_proven_from_estimated]

END FLOW ALGEBRA