Abstract: An $n$-ary operation $Q: Sigma(n) -> Sigma$ is called an $n$-ary quasigroup of order $|\Sigma|$ 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 $Q(x_1, ... , x_n) = P(R(x_{\sigma(1)}, ... , x_{\sigma(k)}), x_{\sigma(k+1)}, ... , x_{\sigma(n)})$ where $P$ and $R$ are $(n-k+1)$-ary and $k$-ary quasigroups, $\sigma$ is a permutation, and $1<k<n$. 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 permutably irreducible $n$-ary quasigroups with permutably reducible $(n-1)$-ary retracts is an open question; however, there are nonexistence results for $5$-ary and $7$-ary quasigroups of order $4$.

Cite: Krotov D.
On irreducible n-ary quasigroups with reducible retracts
European Journal of Combinatorics. 2008. V.29. N2. P.507-513. DOI: 10.1016/j.ejc.2007.01.005 WOS Scopus РИНЦ OpenAlex

Dates:

Submitted:	Oct 26, 2006
Accepted:	Jan 22, 2007
Published online:	Mar 1, 2007

Identifiers:

Web of science:	WOS:000253369500015
Scopus:	2-s2.0-38549094088
Elibrary:	13588561
OpenAlex:	W2039433350

Citing:

DB	Citing
Web of science	6
Scopus	9
OpenAlex	8

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