> 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/_ax.md).

# 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


---

# Agent Instructions
This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com.

## Querying This Documentation
If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question.

Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter:

```
GET https://mandober.gitbook.io/math-debrief/900-index/enums/enum-axioms/axioms-per-theory/_ax.md?ask=<question>&goal=<endgoal>
```

`ask` is the immediate question: it should be specific, self-contained, and written in natural language.
`goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal.

The response will contain a direct answer to the question and relevant excerpts and sources from the documentation.

Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.