Construction and properties of (r,s,t)-inverse quasigroups. I

Keedwell, A. D.; Shcherbacov, Victor

doi:10.1016/s0012-365x(02)00814-2

Cited by 15 publications

(26 citation statements)

References 3 publications

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“…But from the other side, if we apply Cycle to equality (45), then we obtain that e λ−1 x + (y + z) = (e λ−1 x + y) + z (47) for all x, y, z ∈ A. I.e., λ is not a minimal number with declared properties.…”

Section: Begin Cyclementioning

confidence: 97%

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

Shcherbacov¹

2009

J. Gen. 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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Section: Begin Cyclementioning

confidence: 97%

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

Shcherbacov¹

2009

J. Gen. Lie Theory Appl.

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“…However the cross inverse property (xy)I(x) = y holds in an IP-loop only when the loop is commutative. While general m-inverse loops have some common algebraic properties [22], they nevertheless seem rather to be a topic of a combinatorial nature (e.g., see [23] and the subsequent generalizations to quasigroups [24,25]). We shall call a loop a W k IP, k ≥ 1, if it is an m-inverse loop, where m = ((−2) k − 1)/3.…”

Section: Inverse Propertiesmentioning

confidence: 99%

Buchsteiner Loops

Csörgő

Drápal

Kinyon

2009

Int. J. Algebra Comput.

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“…A loop(quasigroup) is an AIPL(AIPQ) if and only if it obeys the identity As observed by Osborn [15], a loop is a WIPL and an AIPL if and only if it is a CIPL. The past efforts of Artzy [2,5,4,3], Belousov and Tzurkan [6] and present studies of Keedwell [11], Keedwell and Shcherbacov [12,13,14] are of great significance in the study of WIPLs, AIPLs, CIPQs and CIPLs, their generalizations(i.e m-inverse loops and quasigroups, (r,s,t)inverse quasigroups) and applications to cryptography.…”

Section: Introductionmentioning

confidence: 99%

A Double Cryptography Using The Keedwell Cross Inverse Quasigroup

Jaiyeola,

Adeniran

2008

Preprint

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The present study further strenghtens the use of the Keedwell CIPQ against attack on a system. This is done as follows. The holomorphic structure of AIPQs(AIPLs) and CIPQs(CIPLs) are investigated. Necessary and sufficient conditions for the holomorph of a quasigroup(loop) to be an AIPQ(AIPL) or CIPQ(CIPL) are established. It is shown that if the holomorph of a quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL), then the holomorph is isomorphic to the quasigroup(loop). Hence, the holomorph of a quasigroup(loop) is an AIPQ(AIPL) or CIPQ(CIPL) if and only if its automorphism group is trivial and the quasigroup(loop) is a AIPQ(AIPL) or CIPQ(CIPL). Furthermore, it is discovered that if the holomorph of a quasigroup(loop) is a CIPQ(CIPL), then the quasigroup(loop) is a flexible unipotent CIPQ(flexible CIPL of exponent 2). By constructing two isotopic quasigroups(loops) U and V such that their automorphism groups are not trivial, it is shown that U is a AIPQ or CIPQ(AIPL or CIPL) if and only if V is a AIPQ or CIPQ(AIPL or CIPL). Explanations and procedures are given on how these CIPQs can be used to double encrypt information.

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Construction and properties of (r,s,t)-inverse quasigroups. I

Cited by 15 publications

References 3 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

Buchsteiner Loops

A Double Cryptography Using The Keedwell Cross Inverse Quasigroup

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