List of Axioms

Binary operations

• ⨀ a binary operation

• ⨁ a binary operation corresponding to addition

• ⨂ a binary operation corresponding to multiplication

In general

• identity:

  • left identity : ϵ ⨀ x = x

  • right identity : x ⨀ ϵ = x

  • total identity : x ⨀ ϵ = x = ϵ ⨀ x

• associativity:

  • left-associative : a ⨀ b ⨀ c ⨀ d = ((a ⨀ b) ⨀ c) ⨀ d

  • right-associative : a ⨀ b ⨀ c ⨀ d = a ⨀ (b ⨀ (c ⨀ d))

  • fully-associative : a ⨀ (b ⨀ c) = (a ⨀ b) ⨀ c = a ⨀ b ⨀ c

  • non-associative

• commutativity : a ⨀ b = b ⨀ a

• idempotency : a ⨀ a = a

• invertibility : a ⨀ a⁻¹ = ϵ = a⁻¹ ⨀ a

• closure, totality : ∀a,∀b ∈ S -> a ⨀ b ∈ S

• distributivity:

  • left distributivity : a ⨂ (b ⨁ c) = (a ⨂ b) ⨁ (a ⨂ c)

  • right distributivity : (b ⨁ c) ⨂ a = (b ⨂ a) ⨁ (c ⨂ a)

  • total distributivity : left and right distributivity

  • when ⨂ is commutative, these 3 axioms are logically equivalent