Related papers: How Associative Can a Non-Associative Moufang Loop…
It is known that with precision till isomorphism that only and only loops $M(F) = M_0(F)/<-1>$, where $M_0(F)$ denotes the loop, consisting from elements of all matrix Cayley-Dickson algebra $C(F)$ with norm 1, and $F$ be a subfield of…
The paper defines the notion of alternative loop algebra F[Q] for any nonassociative Moufang loop Q as being any non-zero homomorphic image of the loop algebra FQ of a loop Q over a field F. For the class M of all nonassociative alternative…
The paper establishes an one-to-one correspondence between simple Moufang loops and Paige loops constructed over Galois extension over prime field in its algebraic closure. Using this connection it describes fully the family of…
Suppose that a binary operation $\circ$ on a finite set $X$ is injective in each variable separately and also associative. It is easy to prove that $(X,\circ)$ must be a group. In this paper we examine what happens if one knows only that a…
It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H ->…
Some physical systems like the quantum mechanics with magnetic charges or field theoretical models appearing in the context of string theory are formulated in terms of non-associative algebras. Hence, demand non-associative star products…
We investigate Moufang loops which can be written as the semidirect product of a loop and a group. We also examine a particular class of loop extensions which arise as a result of a finite cyclic group acting as a group of semiautomorphisms…
Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most…
There are a number of identities which, if satisfied by a Bol loop, imply that the loop is actually Moufang. In this paper we show that in a number of cases, the Moufang identity is also forced not by a single identity, but by giving…
The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We initiate the study of this loop by describing certain Lie and Sabinin algebras related to it.…
In this article we prove that if the additive energy of a strictly increasing sequence $(a_n)$ of natural numbers is less than $N^3/(\log N)^C$ for some $C\geq13.155$, then $(\{a_n\alpha\})$ has Poissonian pair correlation for almost all…
The asymptotic form of the average probability to count $N$ indistinguishable identical particles in a small number $r \ll N$ of binned-together output ports of a $M$-port Haar-random unitary network, proposed recently in \textit{Scientific…
We study the structure of the asymptotic expansion of the probability that a combinatorial object is connected. We show that the coefficients appearing in those asymptotics are integers and can be interpreted as the counting sequences of…
Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ)$ which arises as a result of a construction in \cite{baer}. We investigate some general properties and applications of $\circ$ and determine a necessary and…
A quasigroup $Q$ is called maximally nonassociative if for $x,y,z\in Q$ we have that $x\cdot (y\cdot z) = (x\cdot y)\cdot z$ only if $x=y=z$. We show that, with finitely many exceptions, there exists a maximally nonassociative quasigroup of…
We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting…
A nonassociative generalization of supersymmetry is studied, where supersymmetry generators are considered to be the nonassociative ones. Associators for the product of three and four multipliers are defined. Using a special choice of the…
The Sylow theorems hold for finite extra loops, as does P. Hall's theorem for finite solvable extra loops. Every finite nonassociative extra loop $Q$ has a nontrivial center, $Z(Q)$. Furthermore, $Q/Z(Q)$ is a group whenever $|Q| < 512$.…
In the paper we characterize all interassociates of some non-inverse semigroups and describe up to isomorphism all three-element (strong) doppelsemigroups and their automorphism groups. We prove that there exist $75$ pairwise non-isomorphic…
A question associated with the 2005 open problem of Michael Kinyon (Is every Osborn loop universal?), is answered. Two nice identities that characterize universal (left and right universal) Osborn loops are established. Numerous new…