nLab n-ary quasigroup

Idea

A generalization of quasigroup to n-ary operations.

Definition

Given a natural number $n$, an $n$-ary quasigroup is a set $S$ with an $n$-ary operation $f:S^n \to S$ and $n$ $n$-ary operations $f_i^{-1}:S^n \to S$ for natural numbers $i \lt n$, such that, for $0 \lt m \lt n - 1$,

$$f(f_0^{-1}(a_1, \ldots a_{n - 1}, b), \ldots a_{n - 1}) = b$$

$$f(a_0, \ldots a_{m - 1}, f_m^{-1}(a_0, \ldots a_{m - 1}, a_{m + 1}, \ldots a_{n - 1}, b), a_{m + 1}, \ldots a_{n - 1}) = b$$

$$f(a_0, \ldots a_{n - 2}, f_{n - 1}^{-1}(a_0, \ldots a_{n - 2}, b)) = b$$

See also

• quasigroup

• n-ary magma

• n-ary semigroup

• n-ary group

References

• Wikipedia, n-ary group