# Power-associative magma

From Groupprops

This article defines a property that can be evaluated for a magma, and is invariant under isomorphisms of magmas.

View other such properties

This is a variation of semigroup|Find other variations of semigroup |

## Contents

## Definition

### In terms of the existence of powers

Suppose is a magma, i.e., is a set and is a binary operation on . We say that is power-associative (or equivalently, that is power-associative on ) if we can define a *power function*:

denoted by , such that:

and

.

If such a function exists, it is unique and equals the value of any -fold product of , no matter how it is parenthesized.

### Other definitions

A **power-associative magma** is a magma satisfying the following equivalent conditions:

- It is expressible as the union of subsemigroups, i.e., submagmas that are associative under the operation.
- It is expressible as the union of abelian subsemigroups, i.e., submagmas that are associative and commutative under the operation.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

semigroup | Associativity holds throughout | Diassociative magma|FULL LIST, MORE INFO | ||

diassociative magma | Associativity holds for the submagma generated by two elements | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

magma in which cubes are well-defined | every element commutes with its square | Magma in which cubes and fourth powers are well-defined, Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO | ||

magma in which cubes and fourth powers are well-defined | both make unambiguous sense for all | Magma in which powers up to the fifth are well-defined|FULL LIST, MORE INFO | ||

magma in which powers up to the fifth are well-defined | are well-defined | |FULL LIST, MORE INFO |