In this paper, we give conditions ensuring the existence of a Haar measure in topological IP-loops.

È ben noto che fra le classi di sistemi ternari di Hall (HTS), gli HTS Abeliani ammettano una risoluzione siccome sono esattamente gli spazi affini finiti d'ordine 3; per questi sistemi una tal risoluzione è fornita dalla relazione di parallelismo. In questa nota viene dimostrato che certe classi di HTS non Abeliani costrutti dai gruppi di Burnside $B\left(3,r\right)$, $r\ge 3$ anche ammettono una risoluzione. Allora, questi esempi di HTS si possono considerare anche come spazi finiti di Sperner e dunque la nota conclude...

We investigate the situation when the inner mapping group of a commutative loop is of order $2p$, where $p=4t+3$ is a prime number, and we show that then the loop is solvable.

The algebras ℂ (complex numbers), ℍ (quaternions), and 𝕆 (octonions) are real division algebras obtained from the real numbers ℝ by a doubling procedure called the Cayley-Dickson Process. By doubling ℝ (dim 1), we obtain ℂ (dim 2), then ℂ produces ℍ (dim 4), and ℍ yields 𝕆 (dim 8). The next doubling process applied to 𝕆 then yields an algebra 𝕊 (dim 16) called the sedenions. This study deals with the subalgebra structure of the sedenion algebra 𝕊 and its zero divisors. In particular, it shows...

Loop capable groups are groups which are isomorphic to inner mapping groups of loops. In this paper we show that abelian groups $C_p^k\times C_p\times C_p$, where $k\ge 2$ and $p$ is an odd prime, are not loop capable groups. We also discuss generalizations of this result.

In this paper we consider finite loops and discuss the problem which nilpotent groups are isomorphic to the inner mapping group of a loop. We recall some earlier results and by using connected transversals we transform the problem into a group theoretical one. We will get some new answers as we show that a nilpotent group having either $C_{p^k}\times C_{p^l}$, $k>l\ge 0$ as the Sylow $p$-subgroup for some odd prime $p$ or the group of quaternions as the Sylow $2$-subgroup may not be loop capable.

Information on the structure of finite paramedial quasigroups, including a classification of finite simple paramedial quasigroups, is given. The problem ``Classify the finite simple paramedial quasigroups'' was posed by J. Ježek and T. Kepka at the conference LOOPS'03, Prague 2003.

Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i,j,k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.

The paper deals with quasigroup identities under isotopies. The terminology is taken from [2], [3] and [4]. Stimulated by geometric illustrations, V. D. Belousov in [2] has presented two important identity properties and posed a question for which identities these properties are necessary and sufficient for the identity to be invariant under isotopies. Inspired by V. D. Belousov, G. Monoszová investigated in [6] one special kind of identities for which both Belousov’s properties give necessary and...

We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac{n}{m}\left(\genfrac{}{}{0pt}{}{n}{h}\right)/\left(\genfrac{}{}{0pt}{}{m}{h}\right)$ subsquares of order $m$, where $h=\lceil (m+1)/2\rceil$. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2m}+2$ in $n$. (b) For all $n\ge 5$ there exists a loop of order $n$ in which every...