Index of axioms

• Associativity

  • left-associative, associates to the left, e.g. function application

    e.g. λa.λb.λc.λd.abcd = λa.λb.λc.λd.((ab)c)d

  • right-associative, associates to the right, e.g. function abstraction

    e.g. λa.λb.λc.λd.abcd = λa.(λb.(λc.(λd.abcd)))

  • associative, fully-associative,

    e.g.

    p ∧ q ∧ r ∧ s =

    ((p ∧ q) ∧ r) ∧ s =

    p ∧ (q ∧ (r ∧ s)) =

    (p ∧ q) ∧ (r ∧ s) =

    p ∧ (q ∧ r) ∧ s =

    p ∧ q ∧ r ∧ s

  • non-associative

• Identity

  • total identity

  • left identity

  • right identity

• Invertibility

  • inverse

  • additive inverse

  • multiplicative inverse

• Distributivity

  • total distributivity

  • left distributivity

  • right distributivity

• Commutativity

• Absorption

• Annihilation

• Closure

• Totality

• Domination

• Idempotency

• Complement

Relations

• totality: total relation

• empty relation

• inverse relation

• identity relation

• Reflexivity

  • reflexive relation

  • irreflexive relation

  • coreflexive relation

  • quasi-reflexive relation

• Symmetry

  • symmetric relation

  • asymmetric relation

  • antisymmetric relation

• Transitivity

  • transitive relation

  • antitransitive relation

• Trichotomy

• Linearity

• Euclidean

  • left Euclidean

  • right Euclidean

• Connex

Order

• preorder: asymmetry + transitivity

• equivalence relation: preorder + refexivity

• partiality, partial order: preorder: irreflexive

• totality: total order

• post order

• Well-ordering

Functions

• totality: total function

• partiality: partial function

• Monotonicity

• Involution

  • inverse

• Identity

  • left identity

  • right identity

  • composition, associativity of composition

• Surjection

• Injection

• Bijection

Misc

• Well-formedness

• well-definedness

Type theory

• variance

  • invariance

  • covariance

  • contravariance

Boolean algebra

• complement

• Logic

  • bivalence

  • excluded middle

  • domination

  • De Morgan's laws (distributivity)

  • Principle of explosion

  • completness

• Sets

  • union

  • intersection

  • difference

  • De Morgan's laws (distributivity)

• Algebras

  • closure

  • assoc

  • identity

  • inverse

  • distrib