Editors Belousov's papers and books

A full list of Belousov's papers and books. Back

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W. A. Dudek and V. A. Shcherbacov Remarks to the first publications of V. D. Belousov

It is a short survey and comments to the Belousov's results published in the articles which have not been reviewed neither in Mathematical Reviews nor in Zentralblatt fur Mathematik and are not well-known for the Western mathematicians. Back

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V. D. Belousov Parastrophic-orthogonal quasigroups

Annotated translation of "Parastrophic-orthogonal quasigroups", Acad. Nauk Moldav. SSR, Inst. Mat.s Vychisl. Tsentrom, Kishinev, 1983, prepared by A.D.Keedwell and P.Syrbu based on the original Russian and on an earlier English translation supplied to the first author by Belousov himself. Back

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G. B. Belyavskaya and G. L. Mullen Orthogonal hypercubes and n-ary operations

Connections between orthogonal hypercubes and n-ary operations and quasigroups are studied. Back

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O. Chein and E. Goodaire A new construction of Bol loops: the "odd" case

In "A new construction of Bol loops of order 8k", (J. Algebra 287 (2005), 103-122), the authors presented a new construction of Bol loops as extensions of a loop B by CmxCn, in the case that B contains a central element of order 2 and that m and n are even. In the final section of that paper, the authors remark that the assumption that m and n are even is important and promise to investigate this question in more detail in a future paper. This is that paper. Back

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T. Foguel Amalgam of a loop over an idempotent quasigroup

In this paper we look at away of "gluing" n copies of a loop. This construction gives us a new loop that is a union of n copies of the original loop such that any two of this copies intersect trivially. Back

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A. N. Grishkov, A. V. Zavarnitsine, M. L. Merlini Giuliani Automorphism group of Chein loops

The automorphism group of Chein loops are described. Back

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A. D. Keedwell and V. A. Shcherbacov Quasigroups with an inverse property and generalized parastrophic identities

We study quasigroups (and loops) which have an inverse property. We show that each such quasigroup satisfies a generalized parastrophic identity and that, when investigating properties related to the nuclei, quasigroups which possess any type of inverse property can all be treated in the same way. By means of our approach using autostrophies, we obtain results concerning isomorphisms between or equality of these nuclei. Also, we find conditions for a groupoid which satisfies a generalized parastrophic identity to be a quasigroup. Some of these results generalize our results on (r,s,t)-inverse quasigroups. Back

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V. A. Shcherbacov Finite n-ary medial quasigroups and its automorphism

It is proved that any finite medial n-ary quasigroup is isomorphic to the direct product of a medial unipotently-solvable quasigroup and a principal isotope of a medial idempotent quasigroup. For binary case similar theorem was proved by D. C. Murdoch. This theorem gives a possibility to obtain information on automorphism groups of finite n-ary medial quasigroups. Back

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Z. Stojakovic and W. A. Dudek Conjugate invariant quasigroups

Some properties of conjugate invariant quasigroups and their relations to various combinatorial and algebraic structures are described. Back

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R. Ameri and H. Hedayati Fuzzy isomorphism and quotient of fuzzy subpolygroups

The aim of this note is the study of fuzzy isomorphism and quotient of fuzzy subpolygroups. In this regards first we introduce the notion of fuzzy isomorphism of fuzzy subpolygroups and then we study the quotient of fuzzy subpolygroups. Finally we obtain some related basic results. Back

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S. E. Atani On graded weakly primary ideals

Let G be an arbitrary monoid with identity e. Weakly prime ideals in a commutative ring with non-zero identity have been introduced and studied in 2003 by D. D. Anderson and R. Smith. Here we study the graded weakly primary ideals of a G-graded commutative ring. Various properties of graded weakly primary ideals are considered. For example, we show that an intersection of a family of graded weakly primary ideals such that their homogeneous components are not primary is graded weakly primary. Back

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R. A. Borzooei and H. Harizavi On decomposable hyper BCK-algebras

We introduce the concept of decomposable hyper BCK-algebras and we give a condition for a hyper BCK-algebra to be a decomposable hyper BCK-algebra. Moreover, we state and prove some theorems about (weak, implicative) strong hyper BCK-ideal of a decomposable hyper BCK-algebra. Finally, we give a characterization of some decomposable hyper BCK-algebras. Back

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M. R. Darafsheh Factorization of simple groups involving the alternating group

We find the structure of the finite simple groups G with two subgroups A and B such that G=AB, where A is a non-abelian simple group and B is isomorphic to the alternating group on seven letters. Back

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W. A. Dudek Intuitionistic fuzzy approach to n-ary systems

We adopt the fundamental concepts of intuitionistic fuzzy subalgebras to n-ary groupoids, i.e., on algebras containing one n-ary operation. We describe some similarities and differences between the n-ary and binary case. In the case of n-ary quasigroups and groups we suggest the common method of investigations based on some methods used in the universal algebra. Back

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Z. Kolar--Begovic and V. Volenec Affine regular dodecahedron in GS-quasigroups

The concept of the affine regular dodecahedron is defined in any GS-quasigroup by means of twelve ARP relation which are valid for five out of twenty points. A number of statements about the connection of the corresponding vertices of the dodecahedron will be proved. Quaternary relations Par, GST, DGST can be found in these statements. The theorem of the unique determination of the affine regular dodecahedron by means of its four vertices which satisfy certain conditions will be proved. The geometrical interpretation of all mentioned concepts and relations will be given in some concrete GS-quasigroup. Back

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C. Koscielny Maple implementation of the ElGamal public key encryption scheme working in SMG(pm)

It has been shown that in ElGamal public key encryption scheme working over SMG(pm), the need of finding primitive elements of GF(pm), necessary if the system works traditionally but unfeasible in the case of huge fields, is eliminated. Thus, the discussed system is user-friendly, giving the possibility of very strong encryption with the key of order 10.000 bits and more. The construction of such cryptosystem is of great practical importance, therefore, the paper informs the reader in detail how to resolve this problem using Maple. Back

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C. Koscielny Computing in GF(pm) and in gff(nm) using Maple

It is mentioned in the previous volume that the author intends to show how to construct strong ciphers using SMG(pm). But in order to implement such cryptosystems, an effective tool for computing in GF(pm) and SMG(pm), in the form of an appropriate hardware or software, is needed. The operation of this hardware or software ought to be defined by means of the detailed algorithms. Thus, to get ready the execution of his intention, the author describes in the paper these algorithms, which are represented as routines, written in the comprehensive Maple interpreter, intelligible both for mathematicians and programmers as well. The routines may be used either immediately as elements of encrypting/decrypting procedures in the Maple programming environment or can be easily translated into any compiled programming language (in this case encryption/decryption can be performed at least 100 times faster than in the Maple environment). Aside from that on the basis of the mentioned routines any VLSI chip as the encrypting/decrypting hardware for SMG(pm)- and GF(pm)-based cryptosystems can be produced. It has also been shown that SMG(pm) can be considered as a multiplicative system of an algebraic structure with addition and multiplication operations, containing a large class of systems, including GF(pm). The system is denoted as gff(nm), and multiplication in it is performed modulo an arbitrary polynomial od degree $m$ over the ring Zn. That way gff(nm) is a generalization of Galois field, very well suited for applications in cryptography. This system is named a generalized finite field. Back

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C. Koscielny AES with the increased confidentiality

It has been shown in the paper how to use well known encrypting algorithms AES-128, AES-192 and AES-256 as algorithms AES-340, AES-404 and AES-468, respectively, having considerably increased key space. Back

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V. Krcadinac and V. Volenec A class of quasigroups associated with a cubic Pisot number

In this paper idempotent medial quasigroups satisfying the identity ab.a)a = b are studied. An example are the complex numbers with multiplication defined by a.b = (1-q)a+qb, where q is a solution of q3-2q2+q-1=0. The positive root of this cubic equation can be viewed as a generalization of the golden ratio. It turns out that the quasigroups under consideration have many similar properties to the so-called golden section quasigroups. Back

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M. Polonijo On medial-like identities

The description of the quasigroups that satisfy the identities of the form (ab)(cd) = (p(a)p(b))(p(c)p(d)), where p is a certain permutation on a,b,c,d, is given. Those quasigroups include internally medial (ab)(cd) = (ac)(bd), externally medial (ab)(cd) = (db)(ca) and palindromic (ab)(cd) = (dc)(ba) quasigroups. There are six identities that are the equivalents of commutativity, and fourteen identities are the equivalents of commutative mediality. Back

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