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.

Related notions

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