On irreducible n-ary quasigroups with reducible retracts
Abstract
An n-ary operation q:An->A is called an n-ary quasigroup of order |A| if in x0=q(x1,...,xn) knowledge of any n elements of x0,...,xn uniquely specifies the remaining one. An n-ary quasigroup q is permutably reducible if q(x1,...,xn)=p(r(xs(1),...,xs(k)),xs(k+1),...,xs(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
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.