The authors prove that a local n-quasigroup defined by the equation $x_{n+1}=F(x₁,...,xₙ)=(f₁\left(x₁\right)+...+fₙ\left(xₙ\right))/(x₁+...+xₙ)$, where $f_i\left(x_i\right)$, i,j = 1,...,n, are arbitrary functions, is irreducible if and only if any two functions $f_i\left(x_i\right)$ and $f_j\left(x_j\right)$, i ≠ j, are not both linear homogeneous, or these functions are linear homogeneous but $f_i\left(x_i\right)/x_i\ne f_j\left(x_j\right)/x_j$. This gives a solution of Belousov’s problem to construct examples of irreducible n-quasigroups for any n ≥ 3.

Let $G$ be a division groupoid that is not a quasigroup. For each regular cardinal $\alpha >\left|G\right|$ we construct a quasigroup $Q$ on $G\times \alpha$ that is a quasigroup cover of $G$ (i.e., $G$ is a homomorphic image of $Q$ and $G$ is not an image of any quasigroup that is a proper factor of $Q$). We also show how to easily obtain quasigroup covers from free quasigroups.

We study identities of the form $$L_{x_0}{\varphi }_1\cdots {\varphi }_n{R}_{x_{n+1}}=R_{x_{n+1}}{\varphi }_{\sigma \left(1\right)}\cdots {\varphi }_{\sigma \left(n\right)}{L}_{x_0}$$ in quasigroups, where $n\ge 1$, $\sigma$ is a permutation of $\{1,...,n\}$, and for each $i$, ${\varphi }_i$ is either $L_{x_i}$ or $R_{x_i}$. We prove that in a quasigroup, every such identity implies commutativity. Moreover, if $\sigma$ is chosen randomly and uniformly, it also satisfies associativity with probability approaching $1$ as $n\to \infty$.

It is well known that given a Steiner triple system one can define a quasigroup operation $·$ upon its base set by assigning $x·x=x$ for all $x$ and $x·y=z$, where $z$ is the third point in the block containing the pair $\{x,y\}$. The same can be done for Mendelsohn triple systems, where $(x,y)$ is considered to be ordered. But this is not necessarily the case for directed triple systems. However there do exist directed triple systems, which induce a quasigroup under this operation and these are called Latin directed...

The aim of this paper is to prove that a quasigroup $Q$ with right unit is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by the factor quasigroup $Q/G$ if and only if there exists a normalized left transversal $\Sigma \subset Q$ to $G$ in $Q$ such that the right translations by elements of $\Sigma$ commute with all right translations by elements of the subgroup $G$. Moreover, a loop $Q$ is isomorphic to an $f$-extension of a right nuclear normal subgroup $G$ by a loop if and only if $G$ is middle-nuclear, and...

The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$ $CML$. Any...

This paper completely solves the isomorphism problem for Moufang loops $Q=GC$ where $G⊴Q$ is a noncommutative group with cyclic subgroup of index two and $\left|Z\right(G\left)\right|\le 2$, $C$ is cyclic, $G\cap C=1$, and $Q$ is finite of order coprime to three.

In the present paper we construct the accompanying identity $\widehat{I}$ of a given quasigroup identity $I$. After that we deduce the main result: $I$ is isotopically invariant (i.e., for every guasigroup $Q$ it holds that if $I$ is satisfied in $Q$ then $I$ is satisfied in every quasigroup isotopic to $Q$) if and only if it is equivalent to $\widehat{I}$ (i.e., for every quasigroup $Q$ it holds that in $Q$ either $I,\widehat{I}$ are both satisfied or both not).

Let $G$ be a finite group with a dicyclic subgroup $H$. We show that if there exist $H$-connected transversals in $G$, then $G$ is a solvable group. We apply this result to loop theory and show that if the inner mapping group $I\left(Q\right)$ of a finite loop $Q$ is dicyclic, then $Q$ is a solvable loop. We also discuss a more general solvability criterion in the case where $I\left(Q\right)$ is a certain type of a direct product.

Let $q$ be a positive integer. An algebra is said to have the property $(2,q)$ if all of its subalgebras generated by two distinct elements have exactly $q$ elements. A variety $𝒱$ of algebras is a variety with the property $(2,q)$ if every member of $𝒱$ has the property $(2,q)$. Such varieties exist only in the case of $q$ prime power. By taking the universes of the subalgebras of any finite algebra of a variety with the property $(2,q)$, $2<q$, blocks of Steiner system of type $(2,q)$ are obtained. The stated correspondence...

Let d be a positive integer and μ a generalized Cantor measure satisfying $\mu ={\sum }_{j=1}^m{a}_j\mu \circ S_j^{-1}$, where $0<a_j<1$, ${\sum }_{j=1}^m{a}_j=1$, $S_j=\rho R+b_j$ with 0 < ρ < 1 and R an orthogonal transformation of ${ℝ}^d$. Then ⎧1 < p ≤ 2 ⇒ ⎨$su{p}_{r>0}{r}^{d(1/{\alpha }^{\text{'}}-1/p^{\text{'}})}({\int }_{J_x^r}{\left|\mu ̂\left(y\right)\right|}^{p^{\text{'}}}{dy)}^{1/p^{\text{'}}}\le D₁{\rho }^{-d/{\alpha }^{\text{'}}}$, $x\in {ℝ}^d$, ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’$,$where $J_x^r={\prod }_{i=1}^d(x_i-r/2,x_i+r/2)$, α’ is defined by ${\rho }^{d/{\alpha }^{\text{'}}}={\left({\sum }_{j=1}^m{a}_j^p\right)}^{1/p}$ and the constants D₁ and D₂ depend only on d and p.