The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups
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Bibliographic record
Abstract
Abstract A ternary quasigroup (or 3‐quasigroup) is a pair ( N, q ) where N is an n ‐set and q ( x, y, z ) is a ternary operation on N with unique solvability. A 3‐quasigroup is called 2‐idempotent if it satisfies the generalized idempotent law: q ( x, x, y ) = q ( x, y, x ) = q ( y, x, x )= y . A conjugation of a 3‐quasigroup, considered as an OA (3, 4, n ), \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document} , is a permutation of the coordinate positions applied to the 4‐tuples of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}${\mathcal{B}}$\end{document} . The subgroup of conjugations under which \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document} is invariant is called the conjugate invariant subgroup of \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$({{N}},{\mathcal{B}})$\end{document} . In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n , n ≡7 or 11 (mod 12) and n ≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n ≡0, 1 (mod 3) and n ≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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