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Jonathan D. H. Smith - One of the best experts on this subject based on the ideXlab platform.
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Augmented Quasigroups and character algebras
Advances in Mathematics, 2020Co-Authors: Jonathan D. H. SmithAbstract:Abstract The conjugacy classes of groups and Quasigroups form association schemes, in which the relation products are defined by collapsing group or Quasigroup multiplications. In previous work, sharp transitivity was used to identify association schemes, such as certain Johnson schemes, which cannot appear as Quasigroup schemes. Thus Quasigroup schemes only constitute a fragment of the full set of all association schemes. Nevertheless, the current paper shows that every association scheme is in fact obtained by collapsing a Quasigroup multiplication. In a second application of a similar technique, character Quasigroups are constructed for each finite group, as analogues of the character groups of abelian groups, to encode the multiplicative structure of group characters. As infrastructure for these and related results, three key unifying concepts in compact closed categories are established: augmented comagmas, augmented magmas, and augmented Quasigroups, the latter serving to capture such diverse structures as groups and Heyting algebras.
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Quasigroup words and reversible automata
arXiv: Group Theory, 2019Co-Authors: Jonathan D. H. Smith, Stefanie G WangAbstract:This paper examines two related topics: the linearization of the reversible automata of Gvaramiya and Plotkin, and the problem of finding a faithful representation of the words in a central Quasigroup that respects the triality symmetry of the language of Quasigroups.
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Burnside Orders, Burnside Algebras and Partition Lattices
Order, 2017Co-Authors: Jonathan D. H. SmithAbstract:The permutation representation theory of groups has been extended, through Quasigroups, to one-sided left (or right) Quasigroups. The current paper establishes a link with the theory of ordered sets, introducing the concept of a Burnside order that generalizes the poset of conjugacy classes of subgroups of a finite group. Use of the Burnside order leads to a simplification in the proof of key properties of the Burnside algebra of a left Quasigroup. The Burnside order for a projection left Quasigroup structure on a finite set is defined by the lattice of set partitions of that set, and it is shown that the general direct and restricted tensor product operations for permutation representations of the projection left Quasigroup structure both coincide with the operation of intersection on partitions. In particular, the mark matrix of the Burnside algebra of a projection left Quasigroup, a permutation-theoretic concept, emerges as dual to the zeta function of a partition lattice, an order-theoretic concept.
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Quantum Quasigroups and loops
Journal of Algebra, 2016Co-Authors: Jonathan D. H. SmithAbstract:Abstract Quantum Quasigroups and quantum loops are self-dual objects providing a general framework for the nonassociative extension of quantum group techniques. Bialgebra reducts of Hopf algebras are quantum loops, while sufficient conditions are given for quantum loop structure to augment to a Hopf algebra. The Moufang–Hopf algebras of Benkart et al., the Hopf Quasigroups and coQuasigroups of Klim–Majid, and the coassociative H -bialgebras of Perez-Izquierdo (for instance, the universal enveloping algebras of Sabinin algebras), all form quantum loops. Other quantum Quasigroups offer natural nonassociative extensions of Hopf algebra constructions: Quasigroup algebras, dual Quasigroup algebras, and quantum couples of groups with Quasigroups. Further examples include an algebra of rooted binary trees, and an algebra of skein polynomials.
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Sylow Theory for Quasigroups
Journal of Combinatorial Designs, 2014Co-Authors: Jonathan D. H. SmithAbstract:This paper is intended as a first step toward a general Sylow theory for Quasigroups and Latin squares. A subset of a Quasigroup lies in a nonoverlapping orbit if its respective translates under the elements of the left multiplication group remain disjoint. In the group case, each nonoverlapping orbit contains a subgroup, and Sylow's Theorem guarantees nonoverlapping orbits on subsets whose order is a prime-power divisor of the group order. For the general Quasigroup case, the paper investigates the relationship between non-overlapping orbits and structural properties of a Quasigroup. Divisors of the order of a finite Quasigroup are classified by the behavior of nonoverlapping orbits. In a dual direction, Sylow properties of a subQuasigroup P of a finite left Quasigroup Q may be defined directly in terms of the homogeneous space , and also in terms of the behavior of the isomorphism type within the so-called Burnside order, a labeled order structure on the full set of all isomorphism types of irreducible permutation representations.
Aleksandar Krapež - One of the best experts on this subject based on the ideXlab platform.
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ICT Innovations - Cryptographic Properties of Parastrophic Quasigroup Transformation
ICT Innovations 2012, 2013Co-Authors: Vesna Dimitrova, Verica Bakeva, Aleksandra Popovska-mitrovikj, Aleksandar KrapežAbstract:We consider cryptographic properties of parastrophic Quasigroup transformation defined elsewhere. Using this transformation we classify the Quasigroups of order 4 into three classes: 1) parastrophic fractal; 2) fractal and parastrophic non-fractal; and 3) non-fractal. We investigate the algebraic properties of above classes and present a relationship between fractal and algebraic properties of Quasigroups of order 4. We also find a number of different parastrophes of each Quasigroup of order 4 and use it to divide the set of all Quasigroups of order 4 into four classes. Using these classifications the number of Quasigroups of order 4 which are suitable for designing of cryptographic primitives is increased compared to the case where parastrophes are not used.
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cryptographic properties of parastrophic Quasigroup transformation
International Conference on ICT Innovations, 2012Co-Authors: Vesna Dimitrova, Verica Bakeva, Aleksandra Popovskamitrovikj, Aleksandar KrapežAbstract:We consider cryptographic properties of parastrophic Quasigroup transformation defined elsewhere. Using this transformation we classify the Quasigroups of order 4 into three classes: 1) parastrophic fractal; 2) fractal and parastrophic non-fractal; and 3) non-fractal. We investigate the algebraic properties of above classes and present a relationship between fractal and algebraic properties of Quasigroups of order 4. We also find a number of different parastrophes of each Quasigroup of order 4 and use it to divide the set of all Quasigroups of order 4 into four classes. Using these classifications the number of Quasigroups of order 4 which are suitable for designing of cryptographic primitives is increased compared to the case where parastrophes are not used.
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quadratic level Quasigroup equations with four variables i
Publications De L'institut Mathematique, 2007Co-Authors: Aleksandar KrapežAbstract: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 Forg-Rob and Krapez, and Polonijo. 1. Quasigroups One way to define a Quasigroup is that it is an algebra (S; ·, \ ,/ ) with three binary operations - multiplication (·), left (\ )a nd right (/) division, satisfying the axioms: x\xy = yx (x\y )= y xy/y = x (x/y)y = x. Very often we say that the operation · is a Quasigroup assuming the underlying base set S and the division operations. As usual, whenever unambiguous, the terms like x · y and f (x) are shortened to xy and fx respectively. We review a few basic facts on Quasigroups. More can be found in standard references: Belousov (2), Pflugfelder (20), Chein, Pflugfelder and Smith (6). A loop is a Quasigroup with unit (e), which is a value of constant terms (x\x and y/y) from the additional axiom: (u) x\x = y/y.
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Generalized Associativity on Rectangular Quasigroups
Functional Equations — Results and Advances, 2002Co-Authors: Aleksandar KrapežAbstract:A type of groupoid called a rectangular Quasigroup is defined as a direct product of a left zero semigroup, a Quasigroup and a right zero semigroup. We give three different axiom systems for these groupoids. Some important properties of rectangular Quasigroups are derived, the solvability of the word problem among them.
Vesna Dimitrova - One of the best experts on this subject based on the ideXlab platform.
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pseudo random sequence generators based on the parastrophic Quasigroup transformation
International Conference on ICT Innovations, 2014Co-Authors: Verica Bakeva, Vesna Dimitrova, Mile KostadinoskiAbstract:Pseudo random sequence generators (PRSG) produce sequences of elements that imitate natural random behavior and they have extensive applications in many fields like cryptography, authentication and cryptanalysis. Using Quasigroup string transformations, a PRSG is introduced in [1]. Here, we propose a new design of PRSG using parastrophic Quasigroup transformation defined in [2]. This generator is called Parastrophic Quasigroup Pseudo Random Sequence Generator (PQPRSG). We investigate the goodness of Quasigroups of order 4 for designing of PQPRSQG using classifications given in [3] and linearity of Quasigroups defined in [4]. At the end, we give experimental results about the period of the generator.
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ICT Innovations - Cryptographic Properties of Parastrophic Quasigroup Transformation
ICT Innovations 2012, 2013Co-Authors: Vesna Dimitrova, Verica Bakeva, Aleksandra Popovska-mitrovikj, Aleksandar KrapežAbstract:We consider cryptographic properties of parastrophic Quasigroup transformation defined elsewhere. Using this transformation we classify the Quasigroups of order 4 into three classes: 1) parastrophic fractal; 2) fractal and parastrophic non-fractal; and 3) non-fractal. We investigate the algebraic properties of above classes and present a relationship between fractal and algebraic properties of Quasigroups of order 4. We also find a number of different parastrophes of each Quasigroup of order 4 and use it to divide the set of all Quasigroups of order 4 into four classes. Using these classifications the number of Quasigroups of order 4 which are suitable for designing of cryptographic primitives is increased compared to the case where parastrophes are not used.
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cryptographic properties of parastrophic Quasigroup transformation
International Conference on ICT Innovations, 2012Co-Authors: Vesna Dimitrova, Verica Bakeva, Aleksandra Popovskamitrovikj, Aleksandar KrapežAbstract:We consider cryptographic properties of parastrophic Quasigroup transformation defined elsewhere. Using this transformation we classify the Quasigroups of order 4 into three classes: 1) parastrophic fractal; 2) fractal and parastrophic non-fractal; and 3) non-fractal. We investigate the algebraic properties of above classes and present a relationship between fractal and algebraic properties of Quasigroups of order 4. We also find a number of different parastrophes of each Quasigroup of order 4 and use it to divide the set of all Quasigroups of order 4 into four classes. Using these classifications the number of Quasigroups of order 4 which are suitable for designing of cryptographic primitives is increased compared to the case where parastrophes are not used.
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Periodic Quasigroup string transformations
2009Co-Authors: Vesna Dimitrova, Smile Markovski, Aleksandra MilevaAbstract:Given a finite Quasigroup (Q,*), a Quasigroup string transformations e_l and d_l over the strings of elements from Q are defined as follows. e_l(a_1a_2 ... a_n) = b_1b_2 ... b_n if and only if b_i = b_{i-1}*a_i and d_l(a_1a_2 ... a_n)= b_1b_2 ... b_n if and only if b_i = a_{i-1}*a_i, for each i = 1,2, ... ,n, where l=a_0=b_0 is a fixed element of Q. A Quasigroup string e- or d-transformation t is periodical if for some periodic string we have t(a_1a_2 ... a_k a_1a_2 ... a_k ... a_1a_2 ... a_k) = a_1a_2 ... a_k a_1a_2 ... a_k ... a_1a_2 ... a_k. The Quasigroup string transformations are used in many fields, like: cryptography for designing different cryptographic tools, coding theory for designing error-detecting and error-correcting codes, etc. The properties of the Quasigroup string transformations depend on the used Quasigroups, and some Quasigroups are suitable for cryptographic designs, while some others are suitable for code designs. We give a characterization of the Quasigroups producing periodic string transformations, and for that aim Quasigroups with period k are defined. One can use this characterization for choosing suitable Quasigroups in some applications.
Stipe Vidak - One of the best experts on this subject based on the ideXlab platform.
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Some Theorems of the Euclidean Geometry in Pentagonal Quasigroups
Advances in Intelligent Systems and Computing, 2018Co-Authors: Stipe VidakAbstract: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 \(q^4-3q^3+4q^2-2q+1=0\), some basic geometric concepts are defined in a general pentagonal Quasigroup. Such concepts are parallelogram, midpoint of a segment, regular pentagon and regular decagon with their centres. The connection between pentagonal and, much better known, GS-Quasigroups is mentioned. That connection enables introduction of more geometric concepts in pentagonal Quasigroups. In this article some theorems of the Euclidean geometry which use all these concepts are stated and proved in pentagonal Quasigroups.
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The Napoleon-Barlotti theorem in pentagonal Quasigroups
Glasnik Matematicki, 2016Co-Authors: Stipe VidakAbstract:Pentagonal Quasigroups are IM-Quasigroups in which the additional identity (ab·a)b·a = b holds. GS- Quasigroups are IM-Quasigroups in which the identity a(ab · c) · c = b holds. The relation between these two subclasses of IM-Quasigroups is studied. The geometric concepts of GS-trapezoid and affine regular pentagon, previously defined and studied in GS-Quasigroups, are now defined in a general pentagonal Quasigroup. Along with the concepts of the regular pentagon and the centre of the regular pentagon, previously defined in pentagonal Quasigroups, this enables formulations and proofs of some theorems of the Euclidean plane in a general pentagonal Quasigroup. Among these theorems is the famous Napoleon-Barlotti theorem in the case n = 5.
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Geometry of pentagonal Quasigroups
Publications de l'Institut Math?matique (Belgrade), 2016Co-Authors: Stipe VidakAbstract: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.
Anna A. Taranenko - One of the best experts on this subject based on the ideXlab platform.
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Transversals, near transversals, and diagonals in iterated groups and Quasigroups
arXiv: Combinatorics, 2020Co-Authors: Anna A. TaranenkoAbstract:Given a binary Quasigroup $G$ of order $n$, a $d$-iterated Quasigroup $G[d]$ is the $(d+1)$-ary Quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary Quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary Quasigroups are defined so that to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall--Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general Quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated Quasigroups.
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Transversals in completely reducible multiary Quasigroups and in multiary Quasigroups of order 4
Discrete Mathematics, 2018Co-Authors: Anna A. TaranenkoAbstract:Abstract An n -ary Quasigroup f of order q is an n -ary operation over a set of cardinality q such that the Cayley table of the operation is an n -dimensional latin hypercube of order q . A transversal in a Quasigroup f (or in the corresponding latin hypercube) is a collection of q ( n + 1 ) -tuples from the Cayley table of f , each pair of tuples differing at each position. The problem of transversals in latin hypercubes was posed by Wanless in 2011. An n -ary Quasigroup f is called reducible if it can be obtained as a composition of two Quasigroups whose arity is at least 2, and it is completely reducible if it can be decomposed into binary Quasigroups. In this paper we investigate transversals in reducible Quasigroups and in Quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible Quasigroups. Next we prove that, except for the iterated group Z 4 of even arity, every n -ary Quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in Quasigroups of order 4 and odd arity and count transversals in the iterated group Z 4 of odd arity and in the iterated group Z 2 2 . All results of this paper can be regarded as those concerning latin hypercubes.
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Transversals in completely reducible multiary Quasigroups and in multiary Quasigroups of order 4
arXiv: Combinatorics, 2016Co-Authors: Anna A. TaranenkoAbstract:An $n$-ary Quasigroup $f$ of order $q$ is an $n$-ary operation over a set of cardinality $q$ such that the Cayley table of the operation is an $n$-dimensional latin hypercube of order $q$. A transversal in a Quasigroup $f$ (or in the corresponding latin hypercube) is a collection of $q$ $(n+1)$-tuples belonging to the graph of $f$, each pair of tuples differing at each position. The problem of transversals in multidimensional latin hypercubes was posed by Wanless. A multiary Quasigroup $f$ is called reducible if it can be obtained as a composition of two Quasigroups whose arity is at least 2, and it is completely reducible if it can be decomposed into binary Quasigroups. In this paper we investigate transversals in reducible Quasigroups and in Quasigroups of order 4. We find a lower bound on the number of transversals for a vast class of completely reducible Quasigroups. Next we prove that except the iterated group $\mathbb{Z}_4$ of even arity every $n$-ary Quasigroup of order 4 has a transversal. Also we obtain a lower bound on the number of transversals in Quasigroups of order 4 and odd arity and count the number of transversals in the iterated group $\mathbb{Z}_4$ of odd arity and in the iterated group $\mathbb{Z}_2^2.$ All results of this paper can be regarded as those concerning latin hypercubes.