nLab n-ary quasigroup




A generalization of quasigroup to n-ary operations.


Given a natural number nn, an nn-ary quasigroup is a set SS with an nn-ary operation f:S nSf:S^n \to S and nn nn-ary operations f i 1:S nSf_i^{-1}:S^n \to S for natural numbers i<ni \lt n, such that, for 0<m<n10 \lt m \lt n - 1,

f(f 0 1(a 1,a n1,b),a n1)=bf(f_0^{-1}(a_1, \ldots a_{n - 1}, b), \ldots a_{n - 1}) = b
f(a 0,a m1,f m 1(a 0,a m1,a m+1,a n1,b),a m+1,a n1)=bf(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,a n2,f n1 1(a 0,a n2,b))=bf(a_0, \ldots a_{n - 2}, f_{n - 1}^{-1}(a_0, \ldots a_{n - 2}, b)) = b

See also


Created on August 4, 2022 at 04:39:40. See the history of this page for a list of all contributions to it.