# Index of axioms

* [Associativity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/associativity.md)
  * left-associative
  * right-associative
  * non-associative
* [Identity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/identity.md)
  * total identity
  * left identity
  * right identity
* [Invertibility](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/invertibility.md)
  * inverse
  * additive inverse
  * multiplicative inverse
* [Distributivity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/distributivity.md)
  * total distributivity
  * left distributivity
  * right distributivity
* [Commutativity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/commutativity.md)
  * total commutativity
  * left commutativity
  * right commutativity
* [Absorption](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/absorption.md)
* [Annihilation](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/annihilation.md)
* [Closure](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/closure.md)
* [Totality](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/totality.md)
* [Domination](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/domination.md)
* [Idempotency](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/idempotency.md)
* [Complement](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/complement.md)

**Relations**

* totality: total relation
* empty relation
* inverse relation
* identity relation
* [Reflexivity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/reflexivity.md)
  * reflexive relation
  * irreflexive relation
  * coreflexive relation
  * quasi-reflexive relation
* [Symmetry](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/symmetry.md)
  * symmetric relation
  * asymmetric relation
  * antisymmetric relation
* [Transitivity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/transitivity.md)
  * transitive relation
  * antitransitive relation
* [Trichotomy](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/trichotomy.md)
* [Linearity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/linearity.md)
* [Euclidean](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/euclidean.md)
  * left Euclidean
  * right Euclidean
* [Connex](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/connex.md)

**Order**

* preorder: asymmetry + transitivity
* equivalence relation: preorder + refexivity
* partiality, partial order: preorder: irreflexive
* totality: total order
* post order
* [Well-ordering](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/well-ordering.md)

**Functions**

* totality: total function
* partiality: partial function
* [Monotonicity](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/monotonicity.md)
* [Involution](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/involution.md)
  * inverse
* Identity
  * left identity
  * right identity
  * composition, associativity of composition
* Surjection
* Injection
* Bijection

**Misc**

* [Well-formedness](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/well-formedness.md)
* [well-definedness](https://github.com/mandober/debrief.math/tree/7202acb6ad3d72a6e644cf264793844f5482ff40/100-indices/102-axioms-per-theory/well-definedness.md)

**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


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