Multiary semigroup

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Definition

A multiary semigroup, also called a polyadic semigroup, is a n-ary semigroup for some n \ge 2. Note that the n = 2 case corresponds to the usual notion of semigroup.

A n-ary semigroup is defined as a set G with a n-ary operation, i.e., a map f: G^n \to G such that all different ways of associating expressions involving the n-ary operation f yield equivalent results. Note that this boils down to checking that all the n distinct possible ways of associating an expression of length 2n - 1 give the same answer.

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