Related papers: Parastrophic invariance of Smarandache quasigroups

Every quasigroup $(S,\cdot)$ belongs to a set of 6 quasigroups, called parastrophes denoted by $(S,\pi_i)$, $i\in \{1,2,3,4,5,6\}$. It is shown that isotopy-isomorphy is a necessary and sufficient condition for any two distinct quasigroups…

Parastrophes (conjugates) of a quasigroup can be divided into separate classes containing isotopic parastrophes. We prove that the number of such classes is always 1, 2, 3 or 6. Next we characterize quasigroups having a fixed number of such…

Necessary and sufficient conditions of orthogonality of left (right) linear (alinear) quasigroups in various combinations are given. As corollary we obtain conditions of parastroph orthogonality of left (right) linear (alinear) quasigroups.…

The concept of Smarandache isotopy is introduced and its study is explored for Smarandache: groupoids, quasigroups and loops just like the study of isotopy theory was carried out for groupoids, quasigroups and loops. The exploration…

The class of all quasigroups is covered by six classes: the class of all asymmetric quasigroups and five varieties of quasigroups (commutative, left symmetric, right symmetric, semi-symmetric and totally symmetric). Each of these classes is…

This paper introduced the concept of soft quasigroup, its parastrophes, soft nuclei, left (right) coset, distributive soft quasigroups and normal soft quasigroups. Necessary and sufficient conditions for a soft set over a quasigroup (loop)…

Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where $\chi$ is the extended quadratic character. Let $Q_{a,b}$ be the quasigroup upon…

The isotopic invariance or universality of types and varieties of quasigroups and loops described by one or more equivalent identities has been of interest to researchers in loop theory in the recent past. A variety of quasigroups(loops)…

For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify…

A Smarandache quasigroup(loop) is shown to be universal if all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops…

We construct invariant quasimorphisms for groups acting on the circle. Furthermore, we provide a criterion for the non-extendablity of the resulting quasimorphisms and an explicit formula which relates the values of our quasimorphisms to…

This book has seven chapters. In Chapter one, an elaborate recollection of Smarandache structures like S-semigroups, S-loops, and S-groupoids is given. It also gives notions about N-ary algebraic stuctures and their Smarandache analogue,…

For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary…

It is known that a sequence Pi_i of permutations is quasirandom if and only if the pattern density of every 4-point permutation in Pi_i converges to 1/24. We show that there is a set S of 4-point permutations such that the sum of the…

We prove that PSL(2,Z[1/p]) gives the first example of groups which are not quasi-isometric to each other but have the same quasi-isometry group. Namely, PSL(2,Z[1/p]) and PSL(2,Z[1/q]) are not quasi-isometric unless p=q, and, independent…

The first open case of the Brown, Erd\H{o}s, S\'os conjecture is equivalent to the following; For every $c>0$ there is a threshold $n_0$ so that if a quasigroup has order $n\geq n_0$ then for every subset of triples of the form $(a,b,ab),$…

A quasigroup is a pair $(Q, \cdot)$ where $Q$ is a non-empty set and $\cdot$ is a binary operation on $Q$ such that for every $(u, v) \in Q^2$ there exists a unique $(x, y) \in Q^2$ such that $u \cdot x = v = y \cdot u$. Let $q$ be an odd…

Any pentagonal quasigroup is proved to have the product xy = R(x)+y-R(y) where (Q,+) is an Abelian group, R is its regular automorphism satisfying R^4-R^3+R^2-R+1 = 0 and 1 is the identity mapping. All abelian groups of order n<100 inducing…

Fix $\varepsilon > 0$. We say that a finite group $G$ is $\varepsilon$-quasirandom if every nontrivial irreducible complex representation of $G$ has degree at least $|G|^\varepsilon$. In this paper, we give a structure theorem for large…

We consider $48$ parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non--Belousov (consists of 16 equations) and non--balanced non--gemini…