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Journal articles on the topic 'Quasigroup'

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Published: 4 June 2021

Last updated: 1 February 2022

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1

Khan, Amir, Muhammad Shah, Asif Ali, and Faiz Muhammad. "On Commutative Quasigroup." International Journal of Algebra and Statistics 3, no. 2 (August 22, 2014): 42. http://dx.doi.org/10.20454/ijas.2014.855.

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2

Khudzaifah, Muhammad. "Aplikasi quasigroup dalam pembentukan kunci rahasia pada algoritma hibrida." CAUCHY 3, no. 2 (May 10, 2014): 55. http://dx.doi.org/10.18860/ca.v3i2.2573.

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Pada artikel ini dibahas penerapan quasigrup di bidang kriptografi. Didefinisikan Suatu operasi quasigroup order 𝑝−1 sehingga bisa membentuk suatu algoritma kriptografi yang disebut sebagai quasigrup cipher, quasigrup cipher merupakan algoritma kriptografi simetris. Algoritma kriptografi simetris memiliki sistem keamanan lemah karena kunci yang digunakan untuk proses enciphering sama dengan kunci yang digunakan untuk proses deciphering. Sehingga pada artikel ini algoritma quasigroup cipher dimodifikasi dengan menggabungkannya dengan algoritma RSA menjadi suatu algoritma hibrida yang memiki dua tingkatan kunci

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3

Smith, Jonathan D. H. "Entropy, character theory and centrality of finite quasigroups." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (November 1990): 435–43. http://dx.doi.org/10.1017/s0305004100069334.

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AbstractThe paper introduces concepts of entropy and asymptotic entropy for finite quasigroups. A quasigroup is abelian if and only if its entropy is maximal. It is a З-quasigroup if and only if its asymptotic entropy is maximal.

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4

Kolar-Begović, Zdenka. "An Affine Regular Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup." KoG, no. 21 (2017): 3–5. http://dx.doi.org/10.31896/k.21.8.

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A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satises the identities a\dot (ab \dot c) \dot c = b; a\dot (a \dot bc) \dot c = b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup C(\frac{1}{2} (1 +\sqrt{5})) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

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5

Bennett, F. E. "Quasigroup Identities and Mendelsohn Designs." Canadian Journal of Mathematics 41, no. 2 (April 1, 1989): 341–68. http://dx.doi.org/10.4153/cjm-1989-017-0.

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A quasigroup is an ordered pair (Q, •), where Q is a set and (•) is a binary operation on Q such that the equations ax — b and ya — b are uniquely solvable for every pair of elements a,b in Q. It is well-known (see, for example, [11]) that the multiplication table of a quasigroup defines a Latinsquare, that is, a Latin square can be viewed as the multiplication table of a quasigroup with the headline and sideline removed. We are concerned mainly with finite quasigroups in this paper. A quasigroup (Q, •) is called idempotent if the identity x2 = x holds for all x in Q.

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6

Slaminková, Ivana, and Milan Vojvoda. "Cryptanalysis of a hash function based on isotopy of quasigroups." Tatra Mountains Mathematical Publications 45, no. 1 (December 1, 2010): 137–49. http://dx.doi.org/10.2478/v10127-010-0010-0.

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ABSTRACT This paper deals with cryptanalysis of one hash function based on isotopy of quasigroups [J. Dvorský, E. Ochodková, V. Snášel: Hash functionbased on quasigroups, in: Proc. of Mikulàšska kryptobesídka, Praha, 2001, pp. 27-36. (In Czech)], [J. Dvorský, E. Ochodková, V. Snášel: Hash functionsbased on large quasigroups, in: Proc. of Velikonoční kryptologie, Brno, 2002, pp. 1-8. (In Czech)]. Our work enhances the paper [M. Vojvoda: Cryptanalysisof one hash function based on quasigroup, Tatra Mt. Math. Publ. 29 (2004), 173-181], where the simplified studied hash function, based only on the quasigroup of modular subtraction, was successfully cryptanalysed. In this paper we show how to construct collisions, 2nd preimages, and also preimages for the full hash function based on isotopy of quasigroups.

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7

Dudek, Wieslaw A., and Robert A. R. Monzo. "Pentagonal quasigroups, their translatability and parastrophes." Open Mathematics 19, no. 1 (January 1, 2021): 184–97. http://dx.doi.org/10.1515/math-2021-0004.

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Abstract Any pentagonal quasigroup Q Q is proved to have the product x y = φ ( x ) + y − φ ( y ) xy=\varphi \left(x)+y-\varphi (y) , where ( Q , + ) \left(Q,+) is an Abelian group, φ \varphi is its regular automorphism satisfying φ 4 − φ 3 + φ 2 − φ + ε = 0 {\varphi }^{4}-{\varphi }^{3}+{\varphi }^{2}-\varphi +\varepsilon =0 and ε \varepsilon is the identity mapping. All Abelian groups of order n < 100 n\lt 100 inducing pentagonal quasigroups are determined. The variety of commutative, idempotent, medial groupoids satisfying the pentagonal identity ( x y ⋅ x ) y ⋅ x = y \left(xy\cdot x)y\cdot x=y is proved to be the variety of commutative, pentagonal quasigroups, whose spectrum is { 1 1 n : n = 0 , 1 , 2 , … } \left\{1{1}^{n}:n=0,1,2,\ldots \right\} . We prove that the only translatable commutative pentagonal quasigroup is x y = ( 6 x + 6 y ) ( mod 11 ) xy=\left(6x+6y)\left({\rm{mod}}\hspace{0.33em}11) . The parastrophes of a pentagonal quasigroup are classified according to well-known types of idempotent translatable quasigroups. The translatability of a pentagonal quasigroup induced by the group Z n {{\mathbb{Z}}}_{n} and its automorphism φ ( x ) = a x \varphi \left(x)=ax is proved to determine the value of a a and the range of values of n n .

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8

Shi, Gui-Qi, Xiao-Li Fang, and Blas Torrecillas. "Generalized Yetter–Drinfeld (quasi)modules and Yetter–Drinfeld–Long bi(quasi)modules for Hopf quasigroups." Journal of Algebra and Its Applications 18, no. 02 (February 2019): 1950034. http://dx.doi.org/10.1142/s0219498819500348.

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As generalizations of Yetter–Drinfeld module over a Hopf quasigroup, we introduce the notions of Yetter–Drinfeld–Long bimodule and generalize the Yetter–Drinfeld module over a Hopf quasigroup in this paper, and show that the category of Yetter–Drinfeld–Long bimodules [Formula: see text] over Hopf quasigroups is braided, which generalizes the results in Alonso Álvarez et al. [Projections and Yetter–Drinfeld modules over Hopf (co)quasigroups, J. Algebra 443 (2015) 153–199]. We also prove that the category of [Formula: see text] having all the categories of generalized Yetter–Drinfeld modules [Formula: see text], [Formula: see text] as components is a crossed [Formula: see text]-category.

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9

Volenec, Vladimir, Zdenka Kolar-Begović, and Ružica Kolar-Šuper. "Affine Fullerene C60in a GS-Quasigroup." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/950103.

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It will be shown that the affine fullerene C60, which is defined as an affine image of buckminsterfullerene C60, can be obtained only by means of the golden section. The concept of the affine fullerene C60will be constructed in a general GS-quasigroup using the statements about the relationships between affine regular pentagons and affine regular hexagons. The geometrical interpretation of all discovered relations in a general GS-quasigroup will be given in the GS-quasigroupC(1/2(1+5)).

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10

Damm, H. Michael. "Half quasigroups and generalized quasigroup orthogonality." Discrete Mathematics 311, no. 2-3 (February 2011): 145–53. http://dx.doi.org/10.1016/j.disc.2010.10.004.

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11

Vidak, Stipe. "Geometry of pentagonal quasigroups." Publications de l'Institut Math?matique (Belgrade) 99, no. 113 (2016): 109–20. http://dx.doi.org/10.2298/pim141208013v.

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Pentagonal quasigroups are IM-quasigroups in which the additional identity of pentagonality holds. Motivated by the example C(q), where q is a solution of the equation q4?3q3+4q2?2q+1=0, some basic geometric concepts are introduced and studied in a general pentagonal quasigroup. Such concepts are parallelogram, midpoint of a segment, regular pentagon and regular decagon. Some theorems of Euclidean plane which use these concepts are stated and proved in pentagonal quasigroups.

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12

Im, Bokhee, K. Matczak, and J. D. H. Smith. "Linear diquasigroups." Journal of Algebra and Its Applications 16, no. 03 (March 2017): 1750057. http://dx.doi.org/10.1142/s0219498817500578.

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Following the prototype of dimonoids, diquasigroups are directed versions of quasigroups, where the structure is split into left and right quasigroups on the same set. The linear and affine diquasigroups that form the topic of this paper are built on the foundation of a module. In this context, various issues that may be difficult to handle in the general case, for example identification of the largest two-sided quasigroup image, become more tractable. An appropriate universal algebraic language for affine diquasigroups is established, and the entropic models of this language are characterized. Various interesting classes of linear and affine diquasigroups are singled out for special attention, such as internally associative, Bol, and symmetric diquasigroups. The problem of determining which linear diquasigroups have an abelian group as their undirected replica is raised. One sufficient condition is provided, formulated in terms of a differential calculus for one-sided quasigroup words.

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13

Krapez, Aleksandar. "Weak associativity and quasigroup units." Publications de l'Institut Math?matique (Belgrade) 105, no. 119 (2019): 17–24. http://dx.doi.org/10.2298/pim1919017k.

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We investigate a family of identities similar to weak associativity: x(y/y)?z = x?(y/y)z which might imply the existence of the {left, right, middle} unit in a quasigroup. A partial solution to Krapez, Shcherbacov Problem concerning such identities and consequently to similar well known Belousov's Problem is obtained. Another problem by Krapez and Shcherbacov is solved affirmatively, showing that there are many single identities determining unipotent loops among quasigroups.

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14

Stuhl, Izabella. "Oriented Steiner quasigroups." Journal of Algebra and Its Applications 13, no. 08 (June 24, 2014): 1450072. http://dx.doi.org/10.1142/s0219498814500728.

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We introduce the notion of an oriented Steiner quasigroup and develop elements of a relevant algebraic apparatus. The approach is based upon (modified) Schreier-type f-extensions for quasigroups (cf. earlier works [10, 11, 14]) achieved through oriented Steiner triple systems. This is done in a fashion similar to one in [13] where an analogous construction was established for loops. As a justification of this concept we briefly discuss an application of oriented Steiner triple systems in cryptography using oriented Steiner quasigroups.

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15

Drápal, Aleš, Terry S. Griggs, and Andrew R. Kozlik. "Basics of DTS quasigroups: Algebra, geometry and enumeration." Journal of Algebra and Its Applications 14, no. 06 (April 21, 2015): 1550089. http://dx.doi.org/10.1142/s0219498815500899.

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A directed triple system can be defined as a decomposition of a complete digraph to directed triples 〈x, y, z〉. By setting xy = z, yz = x, xz = y and uu = u we get a binary operation that can be a quasigroup. We give an algebraic description of such quasigroups, explain how they can be associated with triangulated pseudosurfaces and report enumeration results.

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16

Matczak, K., and J. D. H. Smith. "Undirected replicas of directional binary algebras." Journal of Algebra and Its Applications 13, no. 08 (June 24, 2014): 1450051. http://dx.doi.org/10.1142/s0219498814500510.

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Following the prototype of dimonoids, directional algebras are obtained from universal algebras by splitting each fundamental operation into a number of distinct fundamental operations corresponding to directions or selected arguments in the original fundamental operation. Thus dimonoids are directional semigroups, with left- and right-directed multiplications. Directional quasigroups appear in a number of versions, depending on the axiomatization chosen for quasigroups, but this paper concentrates on 4-diquasigroups, which incorporate a left and right quasigroup structure. While introducing several new instances of 4-diquasigroups, including dicores and group-representable diquasigroups, the paper is primarily devoted to the study of undirected replicas of directional binary algebras, dimonoids, digroups, and diquasigroups, where the two directed multiplications are identified. Undirected replicas of diquasigroups are two-sided quasigroups, and thus offer a new approach to the construction of quasigroups of various kinds.

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17

Galatenko, Aleksey V., and Anton E. Pankratiev. "The complexity of checking the polynomial completeness of finite quasigroups." Discrete Mathematics and Applications 30, no. 3 (June 25, 2020): 169–75. http://dx.doi.org/10.1515/dma-2020-0016.

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AbstractThe complexity of the decision of polynomial (functional) completeness of a finite quasigroup is investigated. It is shown that the polynomial completeness of a finite quasigroup may be checked in time polynomially dependent on the order of the quasigroup.

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18

Phillips, J. D., and J. D. H. Smith. "Quasiprimitivity and quasigroups." Bulletin of the Australian Mathematical Society 59, no. 3 (June 1999): 473–75. http://dx.doi.org/10.1017/s0004972700033165.

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It is well known that Q is a simple quasigroup if and only if Mlt Q acts primitively on Q. Here we show that Q is a simple quasigroup if and only if Mlt Q acts quasiprimitively on Q, and that Q is a simple right quasigroup if and only if RMlt Q acts quasiprimitively on Q.

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19

Shahnazaryan, D. A. "ON FUNCTIONAL EQUATION OF ASSOCIATIVITY WITH RIGHT QUASIGROUP OPERATIONS." Proceedings of the YSU A: Physical and Mathematical Sciences 53, no. 3 (250) (December 16, 2019): 150–55. http://dx.doi.org/10.46991/pysu:a/2019.53.3.150.

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We consider equations of associativity and transitivity. We obtain necessary conditions for the solution of these equations for right quasigroup operations, generalizing the classical quasigroup case.

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20

Przytycki, Józef H., and Seung Yeop Yang. "Annihilation of torsion in homology of finite m-AQ quandles." Journal of Knot Theory and Its Ramifications 25, no. 12 (October 2016): 1642012. http://dx.doi.org/10.1142/s0218216516420128.

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It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an [Formula: see text]-almost quasigroup ([Formula: see text]-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.

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21

Smith, Jonathan D. H., and Stefanie G. Wang. "On the enumeration and asymptotic growth of free quasigroup words." International Journal of Algebra and Computation 30, no. 07 (August 8, 2020): 1465–83. http://dx.doi.org/10.1142/s0218196720500496.

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This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

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22

Davidov, Sergey. "On paramedial division groupoids." Asian-European Journal of Mathematics 09, no. 01 (February 22, 2016): 1650008. http://dx.doi.org/10.1142/s179355711650008x.

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In this paper, we prove that regular paramedial division groupoids and binary regular paramedial division algebras are linear over an abelian group. Using this results we prove that every finitely generated paramedial division groupoid is a quasigroup and that every paramedial division groupoid is a homomorphic image of a paramedial quasigroup.

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23

Brožíková, Elena. "On universal quasigroup identities." Mathematica Bohemica 117, no. 1 (1992): 20–32. http://dx.doi.org/10.21136/mb.1992.126237.

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24

Holgate, P. "Logarithmetics and quasigroup structure." Czechoslovak Mathematical Journal 44, no. 2 (1994): 293–304. http://dx.doi.org/10.21136/cmj.1994.128462.

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25

Krapez, Aleksandar. "Quadratic level quasigroup equations with four variables II: The Lattice of varieties." Publications de l'Institut Math?matique (Belgrade) 93, no. 107 (2013): 29–47. http://dx.doi.org/10.2298/pim1307029k.

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We consider a class of quasigroup identities (with one operation symbol) of the form x1x2?x3x4=x5x6?x7x8 and with xi?{x, y, u, v} (1? i ? 8) with each of x, y, u, v occurring exactly twice in the identity. There are 105 such identities. They generate 26 quasigroup varieties. The lattice of these varieties is given.

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26

Krapez, Aleksandar, and Dejan Zivkovic. "Parastrophically equivalent quasigroup equations." Publications de l'Institut Math?matique (Belgrade) 87, no. 101 (2010): 39–58. http://dx.doi.org/10.2298/pim1001039k.

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Fedir M. Sokhats'kyi recently posed four problems concerning parastrophic equivalence between generalized quasigroup functional equations. Sava Krstic in his PhD thesis established a connection between generalized quadratic quasigroup functional equations and connected cubic graphs. We use this connection to solve two of Sokhats'kyi's problems, giving also complete characterization of parastrophic cancellability of quadratic equations and reducing the problem of their classification to the problem of classification of connected cubic graphs. Further, we give formulas for the number of quadratic equations with a given number of variables. Finally, we solve all equations with two variables.

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27

Aleksandar, Krapež, and Marinković Bojan. "Isotopy invariant quasigroup identities." Commentationes Mathematicae Universitatis Carolinae 57, no. 4 (February 20, 2017): 537–47. http://dx.doi.org/10.14712/1213-7243.2015.179.

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28

Krapež, Aleksandar, Slobodan K. Simić, and Dejan V. Tošić. "Parastrophically uncancellable quasigroup equations." Aequationes mathematicae 79, no. 3 (May 13, 2010): 261–80. http://dx.doi.org/10.1007/s00010-010-0016-3.

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29

Phillips, J. D., D. I. Pushkashu, A. V. Shcherbacov, and V. A. Shcherbacov. "On Birkhoff's quasigroup axioms." Journal of Algebra 457 (July 2016): 7–17. http://dx.doi.org/10.1016/j.jalgebra.2016.02.024.

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30

SMITH, JONATHAN D. H. "QUASIGROUP HOMOTOPIES, SEMISYMMETRIZATION, AND REVERSIBLE AUTOMATA." International Journal of Algebra and Computation 18, no. 07 (November 2008): 1203–21. http://dx.doi.org/10.1142/s0218196708004846.

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There have been two distinct approaches to quasigroup homotopies, through reversible automata or through semisymmetrization. In the current paper, these two approaches are correlated. Kernel relations of homotopies are characterized combinatorically, and shown to form a modular lattice. Nets or webs are exhibited as purely algebraic constructs, point sets of objects in the category of quasigroup homotopies. A factorization theorem for morphisms in this category is obtained.

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31

Alonso Álvarez, J. N., J. M. Fernández Vilaboa, and R. González Rodríguez. "Maschke Type Theorems for Weak Hopf Quasigroups." Algebra Colloquium 27, no. 02 (May 7, 2020): 213–30. http://dx.doi.org/10.1142/s1005386720000188.

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In this paper we give necessary and sufficient conditions for a comodule magma over a weak Hopf quasigroup to have a total integral, thus extending the theories developed in the Hopf algebra, weak Hopf algebra and non-associative Hopf algebra contexts. From this result we also deduce a version of Maschke’s theorems for right (H, B)-Hopf triples associated to a weak Hopf quasigroup H and a right H-comodule magma B.

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32

SOLYMOSI, JÓZSEF. "The (7, 4)-Conjecture in Finite Groups." Combinatorics, Probability and Computing 24, no. 4 (February 10, 2015): 680–86. http://dx.doi.org/10.1017/s0963548314000856.

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The first open case of the Brown–Erdős–Sós conjecture is equivalent to the following: for every c > 0, there is a threshold n0 such that if a quasigroup has order n ⩾ n0, then for every subset S of triples of the form (a, b, ab) with |S| ⩾ cn2, there is a seven-element subset of the quasigroup which spans at least four triples of S. In this paper we prove the conjecture for finite groups.

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Sokhatsky, F. M., and A. V. Tarasevych. "Classification of generalized ternary quadratic quasigroup functional equations of the length three." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 179–92. http://dx.doi.org/10.15330/cmp.11.1.179-192.

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A functional equation is called: generalized if all functional variables are pairwise different; ternary if all its functional variables are ternary; quadratic if each individual variable has exactly two appearances; quasigroup if its solutions are studied only on invertible functions. A length of a functional equation is the number of all its functional variables. A complete classification up to parastrophically primary equivalence of generalized quadratic quasigroup functional equations of the length three is given. Solution sets of a full family of representatives of the equivalence are found.

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34

Krapez, Aleksandar. "Quadratic level quasigroup equations with four variables I." Publications de l'Institut Math?matique (Belgrade), no. 95 (2007): 53–67. http://dx.doi.org/10.2298/pim0795053k.

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We consider a class of functional equations with one operational symbol which is assumed to be a quasigroup. Equations are quadratic, level and have four variables each. Therefore, they are of the form x1x2 ? x3x4 = x5x6 ? x7x8 with xi ? {x, y, u, v} (1 _< i _< 8) with each of the variables occurring exactly twice in the equation. There are 105 such equations. They separate into 19 equivalence classes defining 19 quasigroup varieties. The paper (partially) generalizes the results of some recent papers of F?rg-Rob and Krapez, and Polonijo.

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35

Nagy, Péter T., and Izabella Stuhl. "Right Nuclei of Quasigroup Extensions." Communications in Algebra 40, no. 5 (May 2012): 1893–900. http://dx.doi.org/10.1080/00927872.2011.575676.

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36

Battey, Matthew, and Abhishek Parakh. "An Efficient Quasigroup Block Cipher." Wireless Personal Communications 73, no. 1 (December 18, 2012): 63–76. http://dx.doi.org/10.1007/s11277-012-0959-x.

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37

Jedlička, Přemysl. "Examples to Birkhoff's quasigroup axioms." Journal of Algebra 466 (November 2016): 204–7. http://dx.doi.org/10.1016/j.jalgebra.2016.07.029.

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38

Im, Bokhee, and Ji-Young Ryu. "COMPATIBILITY IN CERTAIN QUASIGROUP HOMOGENEOUS SPACE." Bulletin of the Korean Mathematical Society 50, no. 2 (March 31, 2013): 667–74. http://dx.doi.org/10.4134/bkms.2013.50.2.667.

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39

Zaslavsky, Thomas. "Quasigroup associativity and biased expansion graphs." Electronic Research Announcements of the American Mathematical Society 12, no. 2 (February 10, 2006): 13–18. http://dx.doi.org/10.1090/s1079-6762-06-00155-7.

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40

Akivis, Maks A., and Vladislav V. Goldberg. "Local algebras of a differential quasigroup." Bulletin of the American Mathematical Society 43, no. 02 (February 15, 2006): 207–27. http://dx.doi.org/10.1090/s0273-0979-06-01094-9.

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41

Sharma, Monisha. "Generation of quasigroup for cryptographic application." Indian Journal of Science and Technology 2, no. 11 (November 20, 2009): 35–36. http://dx.doi.org/10.17485/ijst/2009/v2i11.11.

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42

Bombardelli, Mea. "Vectors and transfers in hexagonal quasigroup." Glasnik Matematicki 42, no. 2 (December 11, 2007): 363–73. http://dx.doi.org/10.3336/gm.42.2.10.

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43

Im, Bokhee, Ji-Young Ryu, and Jonathan D. H. Smith. "Sharply transitive sets in quasigroup actions." Journal of Algebraic Combinatorics 33, no. 1 (May 19, 2010): 81–93. http://dx.doi.org/10.1007/s10801-010-0234-8.

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44

Smith, Jonathan D. H. "The Burnside algebra of a quasigroup." Journal of Algebra 279, no. 1 (September 2004): 383–401. http://dx.doi.org/10.1016/j.jalgebra.2004.05.004.

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45

Smith, Jonathan D. H. "Quasigroup Homogeneous Spaces and Linear Representations." Journal of Algebra 241, no. 1 (July 2001): 193–203. http://dx.doi.org/10.1006/jabr.2000.8733.

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46

V.A., Kirov. "THREE-BASAL QUASIGROUP WITH GENERALIZED WORDS IDENTITY." Prikladnaya diskretnaya matematika, no. 1 (June 1, 2008): 21–24. http://dx.doi.org/10.17223/20710410/1/4.

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47

Smith, Jonathan D. H. "A coalgebraic approach to quasigroup permutation representations." Algebra Universalis 48, no. 4 (December 1, 2002): 427–38. http://dx.doi.org/10.1007/s000120200010.

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48

Kerby, Brent L., and Jonathan D. H. Smith. "Quasigroup automorphisms and the Norton-Stein complex." Proceedings of the American Mathematical Society 138, no. 09 (September 1, 2010): 3079. http://dx.doi.org/10.1090/s0002-9939-10-10473-0.

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49

Gałuszka, Jan. "Groupoids with quasigroup and Latin square properties." Discrete Mathematics 308, no. 24 (December 2008): 6414–25. http://dx.doi.org/10.1016/j.disc.2007.12.021.

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50

Polin, Sergey V. "Families of quasigroup operations satisfying the generalized distributive law." Discrete Mathematics and Applications 30, no. 3 (June 25, 2020): 187–202. http://dx.doi.org/10.1515/dma-2020-0018.

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Abstract:

AbstractThe previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations.

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