On irreducible n-ary quasigroups with reducible retracts
| dc.creator | Krotov, Denis | |
| dc.date | 2006-07-31 | |
| dc.date | 2007-01-28 | |
| dc.date.accessioned | 2026-07-07T09:37:50Z | |
| dc.date.available | 2026-07-07T09:37:50Z | |
| dc.description | An n-ary operation q:A^n->A is called an n-ary quasigroup of order |A| if in x_0=q(x_1,...,x_n) knowledge of any n elements of x_0,...,x_n uniquely specifies the remaining one. An n-ary quasigroup q is permutably reducible if q(x_1,...,x_n)=p(r(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)}) where p and r are (n-k+1)-ary and k-ary quasigroups, s is a permutation, and 1<k<n. For even n we construct a permutably irreducible n-ary quasigroup of order 4r such that all its retracts obtained by fixing one variable are permutably reducible. We use a partial Boolean function that satisfies similar properties. For odd n the existence of a permutably irreducible n-ary quasigroup such that all its (n-1)-ary retracts are permutably reducible is an open question; however, there are nonexistence results for 5-ary and 7-ary quasigroups of order 4. Keywords:n-ary quasigroups, n-quasigroups, reducibility, Seidel switching, two-graphs | |
| dc.description | 8 p., 1 fig., ACCT-10. v2: revised, the figure improved | |
| dc.identifier | https://arxiv.org/abs/math/0607785 | |
| dc.identifier | http://arxiv.org/abs/math/0607785 | |
| dc.identifier | Eur. J. Comb. 29(2) 2008, 507-513 | |
| dc.identifier | doi:10.1016/j.ejc.2007.01.005 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160600 | |
| dc.subject | Combinatorics | |
| dc.subject | Group Theory | |
| dc.subject | 20N15; 06E30; 05C40 | |
| dc.title | On irreducible n-ary quasigroups with reducible retracts | |
| dc.type | text |