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1.
In this article, we study the spectrum of quasigroups all conjugates of which are distinct and pairwise orthogonal. We call such quasigroups totally conjugate orthogonal quasigroups (for brevity, totCO-quasigroups). Every totCO-quasigroup defines the complete conjugate orthogonal Latin square graph K 6. Examples of totCO-quasigroups of different orders are given. 相似文献
2.
One of the problems in the theory of order polynomially complete lattices is the question whether an order polynomially complete lattice is necessarily finite. In this note we give a partial answer to this problem by showing: No unbounded lattice is order polynomially complete. From this we deduce that a polynomially complete lattice cannot be countably infinite.Presented by I. Rosenberg. 相似文献
3.
Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras. This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper.From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn, and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry. 相似文献
4.
For homotopies of quasigroups, an analog of the fundamental theorem on homomorphisms does not hold in general. In this paper,
we consider two approaches that allow one to obtain an analog of this theorem: the introduction of strict homotopies and the
move from quasigroups to three-sorted quasigroups. 相似文献
5.
Dinamérico P. Pombo 《Rendiconti del Circolo Matematico di Palermo》1921,45(1):119-136
Polynomially bornological, polynomially barrelled and polynomially infrabarrelled locally convex spaces over a non-Archimedean non-trivially valued complete field of characteristic zero are introduced and studied. 相似文献
6.
The computational complexity of shop scheduling problems with multiprocessor tasks on dedicated processors is investigated. The objective is makespan minimization. Preemption of tasks is not allowed. For open and flow-shop problems with three stages, complete classifications into polynomial solvable and NP-hard problems are given. These classifications depend on the compatibility structures of the problems. Furthermore, results for open-shop problems with unit processing times are derived. Finally, it is shown that most of the special cases of the job-shop problem which are polynomially solvable remain polynomially solvable in the multiprocessor task situation.Supported by the Deutsche Forschungsgemeinschaft, Project JoPTAG. 相似文献
7.
There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower bound and a new upper bound on the minimum number of associative triples over quasigroups of a given order. 相似文献
8.
R.C Mullin 《Journal of Combinatorial Theory, Series A》1980,29(3):306-318
The singular direct product employed in the construction of certain designs and quasigroups, is generalized. When applied to skew Room squares, it extends earlier results to show that the spectrum is complete for values above 237. 相似文献
9.
Darryn E. Bryant 《Designs, Codes and Cryptography》1992,2(2):159-168
For m = 6 and for all odd composite integers m, as well as for all even integers m 10 that satisfy certain conditions, 2-perfect m-cycle systems are constructed whose quasigroups have a homomorphism onto quasigroups which do not correspond to a 2-perfect m-cycle systems. Thus it is shown that for these values of m the class of quasigroups arising from all 2-perfect m-cycle systems does not form a variety. 相似文献
10.
Brent Kerby 《Journal of Combinatorial Theory, Series A》2011,118(8):2232-2245
Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T(n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain graph, the cycle graph. The congruence obtained by Norton and Stein extends to the general case, giving a simplified proof (with topology replacing case analysis) of the well-known congruence restriction on the possible orders of general Schroeder quasigroups. Cycle numbers correlate nicely with algebraic properties of quasigroups. Certain well-known classes of quasigroups, such as Schroeder quasigroups and commutative Moufang loops, are shown to maximize the cycle number among all quasigroups belonging to a more general class. 相似文献
11.
12.
Vedran Krčadinac 《Mathematica Slovaca》2011,61(6):885-894
Napoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab·b)(b·ba) = b. In works by V. Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems
of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited
for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids. 相似文献
13.
Jonathan D. H. Smith 《代数通讯》2013,41(5):1878-1885
The article analyzes homotopies between central quasigroups, and their groups of autotopies. In particular, the cycle types of autotopies of central quasigroups and other group isotopes of prime order are identified. 相似文献
14.
A. Krapež 《Aequationes Mathematicae》2001,61(3):255-280
Summary. We define almost quasigroups, a new class of groupoids which generalize quasigroups, and prove several representation theorems for them, essentially reducing them to loops (see Theorems 1, 2 and 9). Some well-known theorems on quasigroups are generalized, notably the theorems of A. A. Albert (Theorems 8, 9 and 10).¶We also define the normal form of equations and show that every generalized linear functional equation Eq on almost quasigroups is equivalent to a system consisting of several equations with at most one variable each, and one equation in the normal form, with the same number of variables as Eq. Eventually, the general solution of the generalized linear functional equations on almost quasigroups with at most two variables is given.¶We plan to solve other generalized linear functional equations in subsequent papers. 相似文献
15.
由-个拟群(Q,(×))可以定义出6个共轭拟群,这6个共轭拟群不一定互不相同,其构成的集合C(Q,(×))的基数t可能的取值是1,2,3或6.记q(n,t)是所有满足|C(Q,(×))|=t的n阶拟群的个数,本文将给出q(n,2)和q(n,6)的计数问题. 相似文献
16.
An algorithm for producing identities which hold in any given finite quasigroup is described. Identities produced by the algorithm are used to prove several results concerning varieties of quasigroups. In particular varieties of quasigroups associated with various combinatorial designs are examined. 相似文献
17.
Margaret E.M. Thomas 《Annals of Pure and Applied Logic》2011,162(6):409
We prove, by explicit construction, that not all sets definable in polynomially bounded o-minimal structures have mild parameterization. Our methods do not depend on the bounds particular to the definition of mildness and therefore our construction is also valid for a generalized form of parameterization, which we call G-mild. Moreover, we present a cell decomposition result for certain o-minimal structures which may be of independent interest. This allows us to show how our construction can produce polynomially bounded, model complete expansions of the real ordered field which, in addition to lacking G-mild parameterization, nonetheless still have analytic cell decomposition. 相似文献
18.
Ronald D. Baker 《Aequationes Mathematicae》1978,18(1-2):296-303
A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schröder's Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Stein's Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Stein's Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups. 相似文献
19.
J.D.H. Smith 《Algebra Universalis》1997,38(2):175-184
The study of quasigroup homotopies reduces to the study of homomorphisms between semisymmetric quasigroups. In particular,
the study of homotopies between central quasigroups reduces to the study of homomorphisms between entropic semisymmetric quasigroups.
Received December 20, 1996; accepted in final form September 17, 1997. 相似文献
20.
Milan Demko 《Czechoslovak Mathematical Journal》2008,58(3):637-650
Some results concerning congruence relations on partially ordered quasigroups (especially, Riesz quasigroups) and ideals of
partially ordered loops are presented. These results generalize the assertions which were proved by Fuchs in [5] for partially
ordered groups and Riesz groups. 相似文献