Cancellative relations and matrices
Canonical Objects in Classes of (n, V)-Groupoids
AMS Subj. Classification: 03C05, 08B20Free algebras are very important in studying classes of algebras, especially varieties of algebras. Any algebra that belongs to a given variety of algebras can be characterized as a homomorphic image of a free algebra of that variety. Describing free algebras is an important task that can be quite complicated, since there is no general method to resolve this problem. The aim of this work is to investigate classes of groupoids, i.e. algebras with one binary operation,...
Cappi di Bruck e loro generalizzazioni
Caratterizzazione di una classe di anelli generalizzati
Categorical models and quasigroup homotopies.
Certain divisible hypergroups.
Certain Functional Equations on Groupoids Weaker than Quasigroups
Characteristic of M-polysymmetrical Hyperrings and some properties of M-Polysymmetrical Hyperrings with Unity
Characterizations of sub-semihypergroups by various triangular norms
We investigate the structure and properties of -sub-semihypergroups, where is an arbitrary triangular norm on a given complete lattice . We study its structure under the direct product and with respect to the fundamental relation. In particular, we consider and , and investigate the connection between -sub-semihypergroups and the probability space.
Characterizing intraregular semigroups by intuitionistic fuzzy sets.
In this paper, we give some theorems which characterize the intraregular semigroups in terms of intuitionistic fuzzy left, right, and biideals.
Characters of finite quasigroups VII: permutation characters
Each homogeneous space of a quasigroup affords a representation of the Bose-Mesner algebra of the association scheme given by the action of the multiplication group. The homogeneous space is said to be faithful if the corresponding representation of the Bose-Mesner algebra is faithful. In the group case, this definition agrees with the usual concept of faithfulness for transitive permutation representations. A permutation character is associated with each quasigroup permutation representation,...
Classification of Bol loops of order 18
Classification of quasigroups according to directions of translations I
It is proved that every translation in a quasigroup has two independent parameters. One of them is a bijection of the carrier set. The second parameter is called a direction here. Properties of directions in a quasigroup are considered in the first part of the work. In particular, totally symmetric, semisymmetric, commutative, left and right symmetric and also asymmetric quasigroups are characterized within these concepts. The sets of translations of the same direction are under consideration in...
Classification of quasigroups according to directions of translations II
In each quasigroup there are defined six types of translations: the left, right and middle translations and their inverses. Two translations may coincide as permutations of , and yet be different when considered upon the web of the quasigroup. We shall call each of the translation types a direction and will associate it with one of the elements and , i.e., the elements of a symmetric group . Properties of the directions are considered in part 1 of “Classification of quasigroups according...
Classification results in quasigroup and loop theory via a combination of automated reasoning tools
We present some novel classification results in quasigroup and loop theory. For quasigroups up to size 5 and loops up to size 7, we describe a unique property which determines the isomorphism (and in the case of loops, the isotopism) class for any example. These invariant properties were generated using a variety of automated techniques --- including machine learning and computer algebra --- which we present here. Moreover, each result has been automatically verified, again using a variety of techniques...
Clifford algebras, Möbius transformations, Vahlen matrices, and -loops
In this paper we show that well-known relationships connecting the Clifford algebra on negative euclidean space, Vahlen matrices, and Möbius transformations extend to connections with the Möbius loop or gyrogroup on the open unit ball in -dimensional euclidean space . One notable achievement is a compact, convenient formula for the Möbius loop operation , where the operations on the right are those arising from the Clifford algebra (a formula comparable to for the Möbius loop multiplication...
Closure conditions of commutativity
There are investigated some closure conditions of Thomsen type in 3-webs which gurantee that at least one of coordinatizing quasigroups of a given 3-web is commutative.
Cohomology groups of a groupoid.
Colimits of representable algebra-valued functors.
Combinatorial aspects of code loops
The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T. Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E. Goodaire contained in [2]. Within this paper, we focus on their combinatorial construction and prove...