> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/900-index/enums/enum-axioms/axioms-per-theory/axioms_all.md).

# 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