# Difference between revisions of "Multiary semigroup"

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(Created page with "==Definition== A '''multiary semigroup''', also called a '''polyadic semigroup''', is a <math>n</math>-ary semigroup for some <math>n \ge 2</math>. Note that the <math>n = 2<...") |
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A <math>n</math>-ary semigroup is defined as a set <math>G</math> with a <math>n</math>-ary operation, i.e., a map <math>f: G^n \to G</math> such that all different ways of associating expressions involving the <math>n</math>-ary operation <math>f</math> yield equivalent results. Note that this boils down to checking that all the <math>n</math> distinct possible ways of associating an expression of length <math>2n - 1</math> give the same answer. | A <math>n</math>-ary semigroup is defined as a set <math>G</math> with a <math>n</math>-ary operation, i.e., a map <math>f: G^n \to G</math> such that all different ways of associating expressions involving the <math>n</math>-ary operation <math>f</math> yield equivalent results. Note that this boils down to checking that all the <math>n</math> distinct possible ways of associating an expression of length <math>2n - 1</math> give the same answer. | ||

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==Related notions== | ==Related notions== |

## Latest revision as of 01:16, 19 June 2012

## Definition

A **multiary semigroup**, also called a **polyadic semigroup**, is a -ary semigroup for some . Note that the case corresponds to the usual notion of semigroup.

A -ary semigroup is defined as a set with a -ary operation, i.e., a map such that all different ways of associating expressions involving the -ary operation yield equivalent results. Note that this boils down to checking that all the distinct possible ways of associating an expression of length give the same answer.

## Related notions

- Multiary quasigroup
- Multiary group is something that is both a multiary semigroup and a multiary quasigroup.