The structure of F-quasigroups

Kepka, Tomáš; Kinyon, Michael; Phillips, J. D.

doi:10.1016/j.jalgebra.2007.05.007

Cited by 14 publications

(40 citation statements)

References 11 publications

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“…, [12], [54,98]) Find necessary and sufficient conditions that a left special loop is isotopic to a left F-quasigroup. Problem 1a has been solved partially by I.A.…”

Section: Problem 1 (Belousov Problem 1amentioning

confidence: 99%

“…In [12] the left special loop is called special. In [49,106,63] left semimedial quasigroups are studied. A quasigroup is trimedial if and only if it is satisfies left and right E-quasigroup equality [63].…”

Section: M-loop If If It Is Left M-and Rightmentioning

confidence: 99%

“…Later Belousov and his pupils I.A. Golovko [12,15,16,40,41,35,33,34,52,86,87,28,54]. In [54,56,57] it is proved that any F-quasigroup is linear over a Moufang loop.…”

Section: Introductionmentioning

confidence: 99%

“…Golovko [12,15,16,40,41,35,33,34,52,86,87,28,54]. In [54,56,57] it is proved that any F-quasigroup is linear over a Moufang loop. The structure of F-quasigroups also is described in [54,56,57].…”

Section: Introductionmentioning

confidence: 99%

“…SMquasigroups are connected with trimedial quasigroups. These quasigroup classes are studied in [48,49,51,106,6,107,62,63]. M. Kinyon and J.D.…”

Section: Introductionmentioning

confidence: 99%

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On the structure of left and right F-, SM-, and E-quasigroups

Shcherbacov¹

2008

J Generalized Lie Theory Appl

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Abstract. It is proved that any left F-quasigroup is isomorphic to the direct product of a left F-quasigroup with a unique idempotent element and isotope of a special form of a left distributive quasigroup. The similar theorems are proved for right F-quasigroups, left and right SM-and E-quasigroups.Information on simple quasigroups from these quasigroup classes is given, for example, finite simple Fquasigroup is a simple group or a simple medial quasigroup.It is proved that any left F-quasigroup is isotopic to the direct product of a group and a left S-loop. Some properties of loop isotopes of F-quasigroups (including M-loops) are pointed out. A left special loop is an isotope of a left F-quasigroup if and only if this loop is isomorphic the direct product of a group and a left S-loop (this is an answer to Belousov "1a" problem).Any left FESM-quasigroup is isotopic to the direct product of an abelian group and a left S-loop (this is an answer to Kinyon-Phillips 2.8(2) problem). New proofs of some known results on the structure of commutative Moufang loops are presented.2000 Mathematics Subject Classification: 20N05

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“…, [12], [54,98]) Find necessary and sufficient conditions that a left special loop is isotopic to a left F-quasigroup. Problem 1a has been solved partially by I.A.…”

Section: Problem 1 (Belousov Problem 1amentioning

confidence: 99%

Section: M-loop If If It Is Left M-and Rightmentioning

confidence: 99%

“…Later Belousov and his pupils I.A. Golovko [12,15,16,40,41,35,33,34,52,86,87,28,54]. In [54,56,57] it is proved that any F-quasigroup is linear over a Moufang loop.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…SMquasigroups are connected with trimedial quasigroups. These quasigroup classes are studied in [48,49,51,106,6,107,62,63]. M. Kinyon and J.D.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the structure of left and right F-, SM-, and E-quasigroups

Shcherbacov¹

2008

J Generalized Lie Theory Appl

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Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction

Kinyon

Veroff

Vojtěchovský

2013

Automated Reasoning and Mathematics

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Abstract. We describe a large-scale project in applied automated deduction concerned with the following problem of considerable interest in loop theory: If Q is a loop with commuting inner mappings, does it follow that Q modulo its center is a group and Q modulo its nucleus is an abelian group? This problem has been answered affirmatively in several varieties of loops. The solution usually involves sophisticated techniques of automated deduction, and the resulting derivations are very long, often with no higher-level human proofs available.

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Distributive and trimedial quasigroups of order 243

Jedlička

Stanovský

Vojtźchovský

2017

Discrete Mathematics

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We enumerate three classes of non-medial quasigroups of order 243 = 3 5 up to isomorphism. There are 17004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Bénéteau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Němec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Opršal and Stanovský).The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the LOOPS package in GAP.2000 Mathematics Subject Classification. Primary: 20N05. Secondary: 05B15, 05B07.

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The structure of F-quasigroups

Cited by 14 publications

References 11 publications

On the structure of left and right F-, SM-, and E-quasigroups

On the structure of left and right F-, SM-, and E-quasigroups

Loops with Abelian Inner Mapping Groups: An Application of Automated Deduction

Distributive and trimedial quasigroups of order 243

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