Multiary semigroup

Definition

A multiary semigroup, also called a polyadic semigroup, is a $n$-ary semigroup for some $n\geq 2$. 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.