• Title/Summary/Keyword: H -quasigroup

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THE GROUP OF STRONG GALOIS OBJECTS ASSOCIATED TO A COCOMMUTATIVE HOPF QUASIGROUP

  • Alvarez, Jose N. Alonso;Rodriguez, Ramon Gonzalez;Vilaboa, Jose M. Fernandez
    • Journal of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.517-543
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    • 2017
  • Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler.

SOME REMARKS ON H𝑣-GROUPS

  • Lee, Dong-Soo;Chung, Sang-Cho
    • Journal of the Chungcheong Mathematical Society
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    • v.14 no.2
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    • pp.9-17
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    • 2001
  • Vogiouklis introduced $H_v$-hyperstructures and gave the "open problem: for $H_v$-groups, we have ${\beta}^*={\beta}^{\prime\prime}$. We have an affirmative result about this open problem for some special cases. We study ${\beta}$ relations on $H_v$-quasigroups. When a set H has at least three elements and (H, ${\cdot}$) is an $H_v$-quasigroup with a weak scalar e, if there are elements $x,y{\in}H$ such that xy = H \ {e}, then we have (xy)(xy) = H.

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MONOIDAL FUNCTORS AND EXACT SEQUENCES OF GROUPS FOR HOPF QUASIGROUPS

  • Alvarez, Jose N. Alonso;Vilaboa, Jose M. Fernandez;Rodriguez, Ramon Gonzalez
    • Journal of the Korean Mathematical Society
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    • v.58 no.2
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    • pp.351-381
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    • 2021
  • In this paper we introduce the notion of strong Galois H-progenerator object for a finite cocommutative Hopf quasigroup H in a symmetric monoidal category C. We prove that the set of isomorphism classes of strong Galois H-progenerator objects is a subgroup of the group of strong Galois H-objects introduced in [3]. Moreover, we show that strong Galois H-progenerator objects are preserved by strong symmetric monoidal functors and, as a consequence, we obtain an exact sequence involving the associated Galois groups. Finally, to the previous functors, if H is finite, we find exact sequences of Picard groups related with invertible left H-(quasi)modules and an isomorphism Pic(HMod) ≅ Pic(C)⊕G(H∗) where Pic(HMod) is the Picard group of the category of left H-modules, Pic(C) the Picard group of C, and G(H∗) the group of group-like morphisms of the dual of H.

$H_{\upsilon}$-GROUPS WITH WEAK SCALAR ELEMENTS

  • Choi, Byung-Mun;Chung, Sang-Cho
    • Journal of applied mathematics & informatics
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    • v.9 no.1
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    • pp.401-413
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    • 2002
  • In this paper, we consider hyperstructures (H,.) defined on the set H = {e, a, b}. We study the hyperstructure of H when every element is one of a scalar unit, a unit or a weak scalar. On those conditions the $H_{\upsilon}$-quasigroups are classified. And we obtain the 15 minimal $H_{\upsilon}$-groups and 2 non-quasi $H_{\upsilon}$-semigroups For these we use the Mathematica 3.0 computer programs.

HOMOGENEOUS CONDITIONS FOR STOCHASTIC TENSORS

  • Im, Bokhee;Smith, Jonathan D.H.
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.371-384
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    • 2022
  • Fix an integer n ≥ 1. Then the simplex Πn, Birkhoff polytope Ωn, and Latin square polytope Λn each yield projective geometries obtained by identifying antipodal points on a sphere bounding a ball centered at the barycenter of the polytope. We investigate conditions for homogeneous coordinates of points in the projective geometries to locate exact vertices of the respective polytopes, namely crisp distributions, permutation matrices, and quasigroups or Latin squares respectively. In the latter case, the homogeneous conditions form a crucial part of a recent projective-geometrical approach to the study of orthogonality of Latin squares. Coordinates based on the barycenter of Ωn are also suited to the analysis of generalized doubly stochastic matrices, observing that orthogonal matrices of this type form a subgroup of the orthogonal group.

COMBINATORIAL SUPERSYMMETRY: SUPERGROUPS, SUPERQUASIGROUPS, AND THEIR MULTIPLICATION GROUPS

  • Bokhee Im;Jonathan D. H. Smith
    • Journal of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.109-132
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    • 2024
  • The Clifford algebra of a direct sum of real quadratic spaces appears as the superalgebra tensor product of the Clifford algebras of the summands. The purpose of the current paper is to present a purely settheoretical version of the superalgebra tensor product which will be applicable equally to groups or to their non-associative analogues - quasigroups and loops. Our work is part of a project to make supersymmetry an effective tool for the study of combinatorial structures. Starting from group and quasigroup structures on four-element supersets, our superproduct unifies the construction of the eight-element quaternion and dihedral groups, further leading to a loop structure which hybridizes the two groups. All three of these loops share the same character table.

ON ALGORITHMS TO COMPUTE SOME Hv-GROUPS

  • Park, Joong-Soo;Chung, Sang-Cho
    • Journal of applied mathematics & informatics
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    • v.7 no.2
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    • pp.553-573
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    • 2000
  • In this paper, we consider hyperstructures (H,·) when H={e,a,b}. We put a condition on (H,·) where e is a unit. We obtain minimal and maximal Hv -groups , semigroups and quasigroups , using Mathematical 3.0 computer programs.

LINEAR AND NON-LINEAR LOOP-TRANSVERSAL CODES IN ERROR-CORRECTION AND GRAPH DOMINATION

  • Dagli, Mehmet;Im, Bokhee;Smith, Jonathan D.H.
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.295-309
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    • 2020
  • Loop transversal codes take an alternative approach to the theory of error-correcting codes, placing emphasis on the set of errors that are to be corrected. Hitherto, the loop transversal code method has been restricted to linear codes. The goal of the current paper is to extend the conceptual framework of loop transversal codes to admit nonlinear codes. We present a natural example of this nonlinearity among perfect single-error correcting codes that exhibit efficient domination in a circulant graph, and contrast it with linear codes in a similar context.