Multiary semigroup

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.