Abstract: An n-ary operation Q : \Sigma^n --> \Sigma is called an n-ary quasigroup of order |\Sigma| if in the equation 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_{\sigma(1)}, ..., x_{\sigma(k)}), x_{\sigma(k+1)}, ..., x_{\sigma(n)}) where P and R are (n-k+1)-ary and k-ary quasigroups, \sigma 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.

Cite: Krotov D.S. , Potapov V.N.
On connection between reducibility of an n-ary quasigroup and that of its retracts
Discrete Mathematics. 2011. V.311. N1. P.58-66. DOI: 10.1016/j.disc.2010.09.023 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Feb 25, 2010
Accepted:	Sep 24, 2010
Published online:	Oct 16, 2010
Published print:	Jan 6, 2011

Identifiers:

Web of science:	WOS:000285172000009
Scopus:	2-s2.0-79953803639
Elibrary:	16999165
OpenAlex:	W2147086976

Citing:

DB	Citing
Web of science	4
Scopus	6
Elibrary	5
OpenAlex	5

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