Narrow Results

In this paper, we propose a new class of error-detecting codes based on quasigroups using automorphisms of the finite field ${𝔽}_{p^n}$ and the additive group $({𝔽}_p,+)$. We demonstrate that these codes are effective in detecting burst errors caused by hardware damage or noisy transmission. We also explore the codes ability to detect various types of double errors, including transposition errors, twin errors, and phonetic errors. Our analysis extends beyond bit-level errors to include block-level errors. Finally, we provide a comparative analysis and demonstrate the real-world applications of our code.

In this paper, we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that, if $(A,H)$ is a matched pair of quasigroupoids, then it is possible to construct a new quasigroupoid $A\bowtie H$ called the double cross product of $A$ and $H$. Also, we show that, if a quasigroupoid $B$ admits an exact factorization, then there exists a matched pair of quasigroupoids $(A,H)$ and an isomorphism of quasigroupoids between $A\bowtie H$ and $B$. Finally, if $𝕂$ is a field, then we show that every matched pair of quasigroupoids $(A,H)$ induces, thanks to the quasigroupoid magma construction, a pair $(𝕂[A],𝕂[H])$ of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup $𝕂[A]\bowtie 𝕂[H]$ isomorphic to $𝕂[A\bowtie H]$ as weak Hopf quasigroups.

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.

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.

The notion of binary representation of algebras with at most two binary operations is introduced in this paper, and the binary version of Cayley theorem for distributive lattices is given by hyperidentities. In particular, we get the binary version of Cayley theorem for DeMorgan and Boolean algebras.

It was shown by van Rees [Subsquares and transversals in latin squares, Ars Combin.29B (1990) 193–204] that a latin square of order $n$ has at most $n^2(n-1)/18$ latin subsquares of order $3$. He conjectured that this bound is only achieved if $n$ is a power of $3$. We show that it can only be achieved if $n\equiv 3mod6$. We also state several conditions that are equivalent to achieving the van Rees bound. One of these is that the Cayley table of a loop achieves the van Rees bound if and only if every loop isotope has exponent $3$. We call such loops van Rees loops and show that they form an equationally defined variety. We also show that: (1) In a van Rees loop, any subloop of index 3 is normal. (2) There are exactly six nonassociative van Rees loops of order $27$ with a nontrivial nucleus and at least 1 with all nuclei trivial. (3) Every commutative van Rees loop has the weak inverse property. (4) For each van Rees loop there is an associated family of Steiner quasigroups.

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.

In this paper, triality refers to the $S_3$-symmetry of the language of quasigroups, which is related to, but distinct from, the notion of triality as the $S_3$-symmetry of the Dynkin diagram $D_4$. The paper investigates a homogeneous method for rendering the linearization of quasigroups (over a commutative ring) naturally invariant under the action of the triality group, on the basis of an appropriate algebra generated by three invertible, non-commuting coefficient variables that is isomorphic to the group algebra of the free group on two generators. The algebra has a natural quotient given by setting the square of each generating variable to be $-1$. The quotient is an algebra of quaternions over the underlying ring, in a way reminiscent of how symmetric groups appear as quotients of braid groups on declaring the generators to be involutions. The corresponding quasigroups (which are described as quaternionic) are characterized by three equivalent pairs of quasigroup identities, permuted by the triality symmetry. The three pairs of identities are logically independent of each other. Totally symmetric quasigroups (such as Steiner triple systems) are quaternionic.

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.

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.

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.

In this paper, the left (right) $h$-nucleus of an invertible algebra is defined and its connection with regular permutations of the invertible algebra is investigated. Using the notions of the $h$-nucleus, we have obtained the characterizations of linear invertible algebras and those for left (right) linear invertible algebras by the second-order formulas.

Large order cryptographically suitable quasigroups have important applications in the development of crypto-primitives and cryptographic schemes. These present new perspectives of cryptography and information security. From algebraic point of view polynomial completeness is one of the most important characteristic for cryptographically suitable quasigroups. In this paper, we propose four different methods to construct polynomially complete quasigroups of any order $n\ge 5$. First method is based on a starting quasigroup of same order, second method is based on a particular permutation of $S{S}_n$ and third and fourth methods are based on products of lower order quasigroups. In the last case, all quasigroups and their isotopes are polynomially complete. We also develop and implement an algorithm to derive a permutation for a given permutation of $S{S}_n,n\ge 5,$ so that they generate whole $S{S}_n$.

In this paper, we extend the result proved by Ulbrich about the characterization of Galois extensions linked to group algebras upon the non-associative (quasigroup) quasigroupoid magma setting. Also, as a particular instance of the results contained in this paper, we obtain the ones proved for Galois extensions related with groupoid algebras.

In this paper, we investigate the properties of A-nuclei and $\sigma$-A-nuclei of a quasigroup including connections between components of $\sigma$-A-nuclei and local identity elements. To make the study easier, we give the explicit structure of $\sigma$-A-nuclei of a loop, a unipotent left loop and a unipotent right loop in terms of translation maps. We also find the left, right and middle A-nuclei of a quasigroup which is an isostrophic image of a loop in terms of translation maps.

In this paper, we investigate the properties of $\sigma$-A-nuclei of a quasigroup including relations between them and relations between their respective component sets, where $\sigma \in S_3$. We also find connections between components of $\sigma$-A-nuclei of a quasigroup and components of $\sigma$-A-nuclei of the isostrophic images. Further, we investigate the properties of various inverse quasigroups using the derived connections. These properties will not only make the study of $\sigma$-A-nuclei of a quasigroup simple but also reduce the time required in the computation of $\sigma$-A-nuclei of a quasigroup for different values of $\sigma$.

Homogeneous systems had their origin in the work of Professor Michihiko Kikkawa. We define a homogeneous system η on a non-empty magma G. Then, η is afterward used to define on G a multiplication μ^(a), where a is a fixed element of G. It was shown that (G, μ^(a)) is a group if and only if η(y, z) ∘ η(x, y) = η(x, z) for all elements x, y, z of G. Let us note that η(x, y) is an application of G into itself for all elements x, y in G. It is our purpose in this paper to find another equivalent condition for which (G, μ^(a)) is a group. And we have obtained η(a, μ^(a)(x, y)) = η(a, x) ∘ η(a, y) for all elements x, y in G.

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.

In this paper we will analyze one linear code from the theoretical point of view. Namely, the code definition is based on linear quasigroups. In the previous work we classified the quasigroups of order 4 according to their probability of undetected errors. Now, in this paper we will conclude whether the linear quasigroups that are in the same class in this classification obtain equal number of surely detected incorrectly transmitted bits. Also, we will classify the linear quasigroups of order 4 according to the number of errors that the code surely detects when they are used for coding. At the end we will make conclusion which quasigroups of order 4 are overall best for coding having in mind both important parameters for every code for error detection: the number of errors that the code surely detects and the probability of undetected errors.

The paper presents an elementary and unified approach to vector spaces over fields of order greater than or equal to one (the latter reducing to sets), based on three key principles. Firstly, use of quasigroups enables the field concept to be redefined in a way that admits a field of order one. Secondly, use of hyperquasigroups provides a recursive definition of linear combination that applies equally well to vector spaces over fields of order greater than or equal to one. Thirdly, it is recognized that relations rather than functions provide the correct morphisms for a category of sets to behave like categories of vector spaces over fields of order greater than one.