On connection between reducibility of an n-ary quasigroup and that of its retracts

详细信息    查看全文

• 作者：Denis S. Krotov ; Vladimir N. Potapov
• 关键词： ; none ; color ; black"" href=""/science?_ob=MathURL&_method=retrieve&_udi=B6V00-517XRRG-2&_mathId=mml31&_user=10&_cdi=5632&_pii=S0012365X10003742&_rdoc=9&_issn=0012365X&_acct=C000050221&_version=1&_userid=10&md5=acd
• 刊名：Discrete Mathematics
• 出版年：2011
• 出版时间：6 January 2011
• 年：2011
• 卷：311
• 期：1
• 页码：58-66
• 全文大小：259 K

文摘

An n-ary operation Q:Σⁿ→Σ is called an n-ary quasigroup of order Σ if in the equation x₀=Q(x₁,…,x_n) knowledge of any n elements of x₀,…,x_n uniquely specifies the remaining one. An n-ary quasigroup Q is (permutably) reducible if Q(x₁,…,x_n)=P(R(x_σ(1),…,x_σ(k)),x_σ(k+1),…,x_σ(n)) where P and R are (n−k+1)-ary and k-ary quasigroups, σ is a permutation, and 1<k<n. An m-ary quasigroup R is called a retract of Q if it can be obtained from Q or one of its inverses by fixing n−m>0 arguments.

We show that every irreducible n-ary quasigroup has an irreducible (n−1)-ary or (n−2)-ary retract; moreover, if the order is finite and prime, then it has an irreducible (n−1)-ary retract. We apply this result to show that all n-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to Z₅ or Z₇ are reducible for n≥4.