Cayley Tables Research Articles

Antiassociative magmas

AbstractA magma S is said to be antiassociative if $$(ab)c \ne a(bc)$$ ( a b ) c ≠ a ( b c ) holds for any elements a, b, c of S. Antiassociative magmas lie at the opposite pole from associative ones, known as semigroups. In this paper, we study antiassociative magmas focusing on their properties and examples that can be seen from Cayley tables. We provide a test for the antiassociativity of a finite magma and give some general methods for constructing antiassociative magmas. We also characterize, describe, and count all antiassociative magma structures on a 3-element set, all their isomorphism classes, and all classes of equivalent magmas of this type.

CLASSIFICATIONS OF UNITARY KRASNER HYPERRINGS OF SMALL ORDER

In this article, we investigate the distributability of the binary operation of monoids with zero compared to the hyperoperation of canonical hypergroups of order 2 and 3with the help of analytical and algebraic methods and without using computer calculations. Then, the unitary Krasner hyperrings of orders 2 and 3 are counted and classified up to isomorphism. In the end, we show the Krasner hyperfields of small order with their Cayley tables.

Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks

This research delves into the extensive analysis of symmetric groups of various orders and degrees. We explore different representations of permutations, construct a Cayley table for the symmetric group of degree four, and systematically identify all its subgroups using Lagrange's theorem. An intriguing discovery unfolds as we demonstrate that the converse of the theorem countered by Sylow's theorem does not hold universally. Furthermore, we apply the concepts of permutations and product of disjoint cycles to address a real-world problem – the Card Trick game. This study amalgamates theoretical exploration with practical applications, showcasing the versatility and depth of symmetric group theory.

A new quasigroup isomorphism invariant arising from fractal image patterns

The analysis and recognition of fractal image patterns derived from Cayley tables has turned out to play a relevant role for distributing distinct types of algebraic and combinatorial structures into isomorphism classes. In this regard, Dimitrova and Markovski described in 2007 a graphical representation of quasigroups by means of fractal image patterns. It is based on the construction of pseudo-random sequences arising from a given quasigroup. In particular, isomorphic quasigroups give rise to the same fractal image pattern, up to permutation of underlying colors. This possible difference may be avoided by homogenizing the standard sets related to these patterns. Based on the differential box-counting method, the mean fractal dimension of homogenized standard sets constitutes a quasigroup isomorphism invariant which is analyzed in this paper in order to distribute quasigroups of the same order into isomorphism classes.

Алгебры бинарных изолирующих формул для теорий сильных произведений

Algebras of distributions of binary isolating and semi-isolating formulas are objects that are derived for a given theory, and they specify the relations between binary formulas of the theory. These algebras are useful for classifying theories and determining which algebras correspond to which theories. In the paper, we discuss algebras of binary formulas for strong products and provide Cayley tables for these algebras. On the basis of constructed tables we formulate a theorem describing all algebras of distributions of binary formulas for the theories of strong multiplications of regular polygons on an edge. In addition, we shows that these algebras can be absorbed by simplex algebras, which simplify the study of that theory and connect it with other algebraic structures. This concept is a useful tool for understanding the relationships between binary formulas of a theory.

Алгебры бинарных изолирующих формул для теорий тензорных произведений

Algebras of distributions of binary isolating and semi-isolating formulae are derived objects for a given theory and reflect binary formula relations between 1-type realizations. These algebras are related to the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to theories from that class, and classify those algebras; 2) classify theories from the class according to the isolating and semi-isolating formulae algebras defined by those theories. The description of a finite algebra of binary isolating formulas unambiguously implies the description of an algebra of binary semi-isolating formulas, which makes it possible to trace the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulas for tensor products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems are formulated describing all algebras of binary formulae distributions for tensor multiplication theory of regular polygons on an edge. It is shown that they are completely described by two algebras

Компьютерное нахождение четырехэлементных мультипликативно идемпотентных полуколец

In this paper we describe a computer program for searching for all four-element multiplicatively idempotent semirings. We have established that up to the isomorphism of such semirings exactly 381, they are represented by Cayley tables of additive and multiplicative reducts. We give the necessary definitions, properties and illustrations.

POINT GROUPS 𝑫𝟐𝒅 and 𝑪𝟑𝒊38T UNDER CLASS SUM APPROACH

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

Completing Partial Transversals of Cayley Tables of Abelian Groups

In 2003 Grüttmüller proved that if $n\geqslant 3$ is odd, then a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $2$ is completable to a transversal. Additionally, he conjectured that a partial transversal of the Cayley table of $\mathbb{Z}_n$ with length $k$ is completable to a transversal if and only if $n$ is odd and either $n \in \{k, k + 1\}$ or $n \geqslant 3k - 1$. Cavenagh, Hämäläinen, and Nelson (in 2009) showed the conjecture is true when $k = 3$ and $n$ is prime. In this paper, we prove Grüttmüller’s conjecture for $k = 2$ and $k = 3$ by establishing a more general result for Cayley tables of Abelian groups of odd order.

FRACTAL MAGMAS AND PUBLIC-KEY CRYPTOGRAPHY

In this paper, we deal with magmas the simplest algebras with a single binary operation. The main result of our research is algorithms for generating chain of finite magmas based on the self-similarity principle of its Cayley tables. In this way the cardinality of a magmas domain is twice as large as the previous one for each magma in the chain, and its Cayley table has a block-like structure. As an example, we consider a cyclic semigroup of binary operations generated by a finite magmas operation with a low-cardinality domain, and a modify the Diffie-Hellman-Merkle key exchange protocol for this case.

Latin squares with maximal partial transversals of many lengths

A partial transversal T of a Latin square L is a set of entries of L in which each row, column and symbol is represented at most once. A partial transversal is maximal if it is not contained in a larger partial transversal. Any maximal partial transversal of a Latin square of order n has size at least ⌈n2⌉ and at most n. We say that a Latin square is omniversal if it possesses a maximal partial transversal of all plausible sizes and is near-omniversal if it possesses a maximal partial transversal of all plausible sizes except one.Evans (2019) showed that omniversal Latin squares of order n exist for any odd n≠3. By extending this result, we show that an omniversal Latin square of order n exists if and only if n∉{3,4} and n≢2(mod4). Furthermore, we show that near-omniversal Latin squares exist for all orders n≡2(mod4).Finally, we show that no non-trivial finite group has an omniversal Cayley table, and only 15 finite groups have a near-omniversal Cayley table. In fact, as n grows, Cayley tables of groups of order n miss a constant fraction of the plausible sizes of maximal partial transversals. In the course of proving this, we partially solve the following interesting problem in combinatorial group theory. Suppose that we have two finite subsets R,C⊆G of a group G such that |{rc:r∈R,c∈C}|=m. How large do |R| and |C| need to be (in terms of m) to be certain that R⊆xH and C⊆Hy for some subgroup H of order m in G, and x,y∈G?

Алгебры бинарных изолирующих формул для теорий корневых произведений графов

Algebras of distributions of binary isolating and semi-isolating formulas are derived objects for given theory and reflect binary formula relations between realizations of 1-types. These algebras are associated with the following natural classification questions: 1) for a given class of theories, determine which algebras correspond to the theories from this class and classify these algebras; 2) to classify theories from a given class depending on the algebras defined by these theories of isolating and semi-isolating formulas. Here the description of a finite algebra of binary isolating formulas unambiguously entails a description of the algebra of binary semi-isolating formulas, which makes it possible to track the behavior of all binary formula relations of a given theory. The paper describes algebras of binary formulae for root products. The Cayley tables are given for the obtained algebras. Based on these tables, theorems describing all algebras of binary formulae distributions for the root multiplication theory of regular polygons on an edge are formulated. It is shown that they are completely described by two algebras.

Nearly Linear Time Isomorphism Algorithms for Some Nonabelian Group Classes

The isomorphism problem for groups, when the groups are given by their Cayley tables is a well-studied problem. This problem has been studied for various restricted classes of groups. Kavitha gave a linear time isomorphism algorithm for abelian groups (JCSS 2007). Although there are isomorphism algorithms for certain nonabelian group classes represented by their Cayley tables, the complexities of those algorithms are usually super-linear. In this paper, we design linear and nearly linear time isomorphism algorithms for some nonabelian groups. More precisely,

Latin Squares over Quasigroups

We propose a construction that allows generating large families of Latin squares, i.e., Cayley tables of finite quasigroups. This construction generalizes proper families of functions over Abelian groups introduced by Nosov and Pankratiev. We also show that all quasigroups generated by the original construction contain at least one subquasigroup, while the generalized construction generates quasigroups free of subquasigroups.

Workshop Penggunaan Media Pembelajaran Berbasis Sampah dan Barang Bekas Bagi Guru MIN TTU

Permasalahan pokok di Madrasah Ibtidayah Negeri (MIN) TTU-Kefamenanu yang menjadi mitra pengabdian ini adalah: (i) belum tersedianya alat peraga matematika di sekolah mitra; (ii) belum pernah diselenggarakan kegiatan workshop tentang penggunaan alat peraga matematika di sekolah mitra. Metode pelaksanaan pengabdian ini adalah ceramah, demonstrasi, praktek langsung, diskusi dan penugasan. Adapun alat peraga yang digunakan antara lain: Mistar Bilangan, Kartu Positif Negatif, Puzzle FPB dan KPK, Magic Square, Perkalian Kanada, Congklak Penjumlahan, Tabel Cayley, Ular Tangga Aljabar, Menara Hanoi dan Tangram. Alat peraga matematika ini terbuat dari sampah dan barang bekas. Pelaksanaan pengabdian selama 3 hari dengan rincian satu hari workshop dengan peserta para guru dan dua hari praktek penggunaan alat peraga dengan peserta siswa kelas IV, V dan VI. Hasil evaluasi pengabdian menunjukkan bahwa i) 33,33% peserta sangat setuju dan 66,67% peserta workshop setuju bahwa materi workshop terorganisasi dengan baik dan alat peraga yang digunakan sesuai dengan materi pembelajaran di sekolah dasar, alat peraga menarik dan membantu siswa dalam belajar sambil bermain; ii) 80% peserta setuju dan 20% sangat setuju bahwa panduan praktik penggunaan alat peraga dengan LKS yang disediakan dapat dimengerti dengan baik.

Lagrange's theorem for Hom-Groups

Hom-groups are nonassociative generalizations of groups where the unitality and associativity are twisted by a map. We show that a Hom-group $(G, \alpha )$ is a pointed idempotent quasigroup (pique). We use Cayley tables of quasigroups to introduce some examples of Hom-groups. Introducing the notions of Hom-subgroups and cosets we prove Lagrange's theorem for finite Hom-groups. This states that the order of any Hom-subgroup $H$ of a finite Hom-group $G$ divides the order of $G$. We linearize Hom-groups to obtain a class of nonassociative Hopf algebras called Hom-Hopf algebras. As an application of our results, we show that the dimension of a Hom-sub-Hopf algebra of the finite dimensional Hom-group Hopf algebra $\mathbb {K}G$ divides the order of $G$. The new tools introduced in this paper could potentially have applications in theories of quasigroups, nonassociative Hopf algebras, Hom-type objects, combinatorics, and cryptography.

The Chromatic Number of Finite Group Cayley Tables

The chromatic number of a latin square $L$, denoted $\chi(L)$, is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies $\chi(L) \leq |L|+2$. If true, this would resolve a longstanding conjecture—commonly attributed to Brualdi—that every latin square has a partial transversal of size $|L|-1$. Restricting our attention to Cayley tables of finite groups, we prove two results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group $G$ has chromatic number $|G|$ or $|G|+2$, with the latter case occurring if and only if $G$ has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For $|G|\geq 3$, this improves the best-known general upper bound from $2|G|$ to $\frac{3}{2}|G|$, while yielding an even stronger result in infinitely many cases.

Polynomially Complete Quasigroups of Prime Order

We formulate a polynomial completeness criterion for quasigroups of prime order, and show that verification of polynomial completeness may require time polynomial in order. The results obtained are generalized to n-quasigroups for any n ≥ 3. In conclusion, simple corollaries are given on the share of polynomially complete quasigroups among all quasigroups, and on the cycle structure of row and column permutations in Cayley tables for quasigroups that are not polynomially complete.

On Algebras of Distributions of Binary Isolating Formulas for Theories of Abelian Groups and Their Ordered Enrichments

We describe algebras of distributions of binary isolating formulas for theories of abelian groups and some of their ordered enrichments. The base of this description is the general theory of algebras of isolating formulas. We also take into account the specificity of the basedness of theories of abelian groups on Szmielew invariants. We give Cayley tables for algebras that correspond to theories of basic abelian groups and their ordered enrichments and propose a technique for transforming algebras for theories of basic abelian groups into algebras for arbitrary theories of abelian groups.

Visualizing large-order groups with computer-generated Cayley tables

ABSTRACTLarge-order finite groups have been made the subject of artwork through abstract computer-generated pictures corresponding to Cayley tables which have been presented in some recent mathematical art exhibits and galleries. In our present article, we further explore the concept of using computer art to visualize large-order groups, by offering some new computer-generated large-scale coloured Cayley tables. We present artistic depictions of basic group-theoretic concepts, and we also discuss some potential pedagogical benefits of computer-generated artwork based on large-scale Cayley tables. We also consider some open questions concerning group theory and pedagogy in mathematics.