template.qmd

---
title: Axioms
terms:
  - term: Natural number
    regex: "[Nn]atural numbers?"
    associatedMacros:
    - command: "N"
      description: "The Natural Numbers: $1, 2, 3, ...$"
      macro: '\\mathbb{N}'
  - term: associativity
    regex: "[Aa]ssociativ[ei]t?y?"
    id: def-associativity
  - term: identity
    regex: "[Ii]dentit[yi]e?s?"
    id: def-identity
  - term: inverse
    regex: "[Ii]nverses?"
    id: def-inverse
macros:
  - command: Inverse
    macro: '{#1}^{-1}'
    variables: 1
  - command: Set
    macro: '{S}'
  - command: Operation
    macro: '{#1}\\cdot{#2}'
    variables: 2
  - command: Identity
    macro: '{e}'
author: Kai Prince SFHEA
date: last-modified
---

# Introduction

::: {#def-associativity templateMap="[\\Set,\\Operation,\\Identity]"}
## associativity

For all ${#1},{#2},{#3}$ in $\Set$, one has $\Operation{(\Operation{#1}{#2})}{#3}=\Operation{#1}{(\Operation{#2}{#3})}$.
:::

::: {#def-identity templateMap="[\\Set,\\Operation,\\Identity]"}
## identity
There exists an element $\Identity$ in $\Set$ such that, for every ${#1}$ in $\Set$, one has $\Operation{\Identity}{#1}={#1}$ and $\Operation{#1}{\Identity}={#1}$. Such an element is unique and is called the *identity element*.
:::

:::: {#def-inverse templateMap="[\\Set,\\Operation,\\Identity]"}
## unique inverse
For each ${#1}$ in $\Set$, there exists an element ${#2}$ in $\Set$ such that $\Operation{#1}{#2}=\Identity$ and $\Operation{#2}{#1}=\Identity$, where $\Identity$ is the identity element.

For each ${#1}$, the element ${#2}$ is unique and is called the *inverse* of ${#2}$ and is denoted $\Inverse{#1}$.
::::

Test @def-associativity

# More Information

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