# Magma

From Groupprops

## Contents

## Definition

A set equipped with a binary operation is termed a **magma**.

Magmas are also sometimes termed *groupoids*, but the term groupoid has another more common use.

## Relation with other structures

### Additional structures and conditions

The following are the common choices of additional conditions usually imposed on the binary operation of a magma:

- The existence of a neutral element, i.e., an identity element for the binary operation. For a binary operation , a left neutral element is an element such that for all and a right neutral element is an element such that for all . A (two-sided) neutral element is an element that is both left and right neutral.
- The associativity condition, which states that for all in the underlying set of the magma. There are many weaker versions of associativity, which require only certain kinds of expressions to
*associate*. - The commutativity condition, which states that for all in the underlying set of the magma. There are weaker versions of commutativity, which requre only certain kinds of expressions to
*commute*. - The existence of a nil element, i.e., a zero element for the binary operation. is a left nil for if for all , and a right nil for if for all .
- The existence of inverse elements with respect to a two-sided neutral element. A left inverse for with respect to is a such that , and a right inverse is a such that .
- Cancellation. An element is left-cancellative if and right-cancellative if .
- Existence of solutions to
*left/right quotient*equations: Whether an equation of the form can be solved, for given . Similarly, whether can be solved for given .

### Terminology for magmas with these additional conditions

A **Yes** means that the structure/condition is *always* true for that type of magma while a **No** means that it is not necessarily true. The collection of Yeses in any row completely defines the type of magma being considered.

Name | Two-sided neutral element | Associativity | Commutativity | Inverses | Cancellation | Existence of quotients |
---|---|---|---|---|---|---|

Magma | No | No | No | No | No | No |

Unital magma | Yes | No | No | No | No | No |

Cancellative magma | No | No | No | No | Yes | No |

Semigroup | No | Yes | No | No | No | No |

Cancellative semigroup | No | Yes | No | No | Yes | No |

Monoid | Yes | Yes | No | No | No | No |

Quasigroup | No | No | No | No | Yes | Yes |

Algebra loop | Yes | No | No | Yes | Yes | Yes |

Group | Yes | Yes | No | Yes | Yes | Yes |

Abelian group | Yes | Yes | Yes | Yes | Yes | Yes |