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Steiner loops of affine type are associated to arbitrary Steiner triple systems. They behave to elementary abelian 3-groups as arbitrary Steiner Triple Systems behave to affine geometries over GF (3) . We investigate algebraic and geometric properties of these loops often in connection to configurations. Steiner loops of affine type, as extensions of normal subloops by factor loops, are studied. We prove that the multiplication group of every Steiner loop of affine type with n elements is contained in the alternating group A n and we give conditions for those loops having A n as their multiplication groups (and hence for the loops being simple). [ABSTRACT FROM AUTHOR]

In this paper, implementation of stream cipher algorithm based on nonlinear combination generator for any given sequences is introduced. The main contribution of this work is to enhance the security of this generator and level of complexity is added using logic functions. The proposed scheme was measured through a series of randomness tests. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the theory of multiplication alteration by two-cocycles for non-associative structures like non-associative bimonoids with left (right) division. Also, we explore the connections between Yetter–Drinfeld modules for Hopf quasigroups, projections of Hopf quasigroups, skew pairings and quasitriangular structures, obtaining the non-associative version of the main results proved by Doi and Takeuchi in the Hopf algebra setting. [ABSTRACT FROM AUTHOR]

A division sudoku is a latin square whose all six conjugates are sudoku squares. We enumerate division sudokus up to a suitable equivalence, introduce powerful invariants of division sudokus, and also study latin squares that are division sudokus with respect to multiple partitions at the same time. We use nearfields and affine geometry to construct division sudokus of prime power rank that are rich in sudoku partitions. [ABSTRACT FROM AUTHOR]

We show that for all $m,k,r\in \mathbb{N}$ , there is an $n\in \mathbb{N}$ such that whenever $L$ is a Latin square of order $m$ and the Cartesian product $L^{n}$ of $n$ copies of $L$ is $r$ -coloured, there is a monochrome Latin subsquare of $L^{n}$ , isotopic to $L^{k}$. In particular, for every prime $p$ and for all $k,r\in \mathbb{N}$ , there is an $n\in \mathbb{N}$ such that whenever the multiplication table $L({\mathbb{Z}_{p}}^{n})$ of the group ${\mathbb{Z}_{p}}^{n}$ is $r$ -coloured, there is a monochrome Latin subsquare of order $p^{k}$. On the other hand, we show that for every group $G$ of order $\leq 15$ , there is a 2-colouring of $L(G)$ without a nontrivial monochrome Latin subsquare. [ABSTRACT FROM AUTHOR]

This paper is dedicated to Karl Strambach on the occasion of his 80th birthday. Here we want to describe our work with Prof. Karl Strambach. [ABSTRACT FROM AUTHOR]

Residual finiteness is known to be an important property of groups appearing in combinatorial group theory and low dimensional topology. In a recent work (Bardakov et al. in Proc Am Math Soc 147:3621–3633, 2019. 10.1090/proc/14488) residual finiteness of quandles was introduced, and it was proved that free quandles and knot quandles are residually finite. In this paper, we extend these results and prove that free products of residually finite quandles are residually finite provided their associated groups are residually finite. As associated groups of link quandles are link groups, which are known to be residually finite, it follows that link quandles are residually finite. [ABSTRACT FROM AUTHOR]

Automorphic loops, or A-loops, are loops in which all inner mappings are automorphisms. Here, we investigate a class of A-loops known as the dihedral automorphic loop, denoted by D i h (α , G). This class is constructed from a finite abelian group G and an automorphism of G, generalizing the construction of the dihedral group which has order 2 | G |. A half-isomorphism of loops is a bijection f between loops L , L ′ where for any x , y ∈ L , we have f (x y) ∈ { f (x) f (y) , f (y) f (x) }. We say that a half-isomorphism is nontrivial when it is neither an isomorphism nor an anti-isomorphism. In this paper, we prove that D i h (α , G) has nontrivial half-isomorphisms and we classify nontrivial half-isomorphisms between dihedral automorphic loops. Also we identify the group of half-automorphisms of this class of A-loops. [ABSTRACT FROM AUTHOR]

A loop X is said to satisfy Moufang's theorem if for every x , y , z ∈ X such that x (y z) = (x y) z the subloop generated by x, y, z is a group. We prove that the variety V of Steiner loops satisfying the identity (x z) (((x y) z) (y z)) = ((x z) ((x y) z)) (y z) is not contained in the variety of Moufang loops, yet every loop in V satisfies Moufang's theorem. This solves a problem posed by Andrew Rajah. [ABSTRACT FROM AUTHOR]

In this article, we show that any finite gyrogroup can be represented on a space of complex-valued functions. In particular, we prove that any linear representation of a finite gyrogroup on a finite-dimensional complex inner product space is unitary and hence is completely reducible using strong connections between linear actions of groups and gyrogroups. Also, we provide an example of a unitary representation of an arbitrary finite gyrogroup, which resembles the group-theoretic left regular representation. [ABSTRACT FROM AUTHOR]

A loop Q satisfies the Moufang theorem if Q possesses the following property: let x , y , z ∈ Q be associate, that is x (y z) = (x y) z , then x, y, z generate a subgroup of Q. It remains an open problem in the theory of loops to find a non-Moufang variety of loops satisfying the Moufang Theorem. We describe the non-Moufang varieties Bn of diassociative loops such that all analytic loops from Bn satisfy the above mentioned property. [ABSTRACT FROM AUTHOR]

In this paper the notions of solvable right transversal and nilpotent right transversal are defined. Further, it is proved that if a core-free subgroup has a generating solvable transversal, then the whole group is solvable.

In this paper, we introduce the notion of weak nonassociative Doi–Hopf module and give the Fundamental Theorem of Hopf modules in this setting. Also, we prove that there exists a categorical equivalence that admits as particular instances the ones constructed in the literature for Hopf algebras, weak Hopf algebras, Hopf quasigroups and weak Hopf quasigroups. [ABSTRACT FROM AUTHOR]

Latin squares are used as scramblers on symmetric-key algorithms that generate pseudo-random sequences of the same length. The robustness and effectiveness of these algorithms are respectively based on the extremely large key space and the appropriate choice of the Latin square under consideration. It is also known the importance that isomorphism classes of Latin squares have to design an effective algorithm. In order to delve into this last aspect, we improve in this paper the efficiency of the known methods on computational algebraic geometry to enumerate and classify partial Latin squares. Particularly, we introduce the notion of affine algebraic set of a partial Latin square L = (lij) of order n over a field K as the set of zeros of the binomial ideal 〈 x i x j − x l ij : (i , j) is a non-empty cell in L 〉 ⊆ K [ x 1 , ... , x n ] . Since isomorphic partial Latin squares give rise to isomorphic affine algebraic sets, every isomorphism invariant of the latter constitutes an isomorphism invariant of the former. In particular, we deal computationally with the problem of deciding whether two given partial Latin squares have either the same or isomorphic affine algebraic sets. To this end, we introduce a new pair of equivalence relations among partial Latin squares: being partial transpose and being partial isotopic. [ABSTRACT FROM AUTHOR]

The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer n ≥ 2 , the nth-commutativity degree of a finite algebraic structure S, denoted by P n (S) , is the probability that for chosen randomly two elements x and y of S, the relator x n y = y x n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability. [ABSTRACT FROM AUTHOR]

In this paper we generalize Belousov's theorem on linearity of invertible algebras with ∀∃(∀) identities. The obtained results can be used for classification of medial hyperidentities. [ABSTRACT FROM AUTHOR]

It has been proven that all Moufang loops of order p4q3 where p and q are odd primes with 3 are associative if and only if q≢1(modp). Also all Moufang loops of order p13p23⋯pn3 where each pi are a prime with 3≤p1 were proven to be associative if and only if pi≢1(modpj). In this paper, we extend these results to Moufang loops of order p4q13q23⋯qn3 where p and each qi is a prime with 3 and qi≢1(modqj). [ABSTRACT FROM AUTHOR]

A compact, Hausdorff and totally disconnected loop is called a boolean loop . In this note, we prove that a boolean loop with compact left inner mapping group is profinite. [ABSTRACT FROM AUTHOR]

By means of the method of transversals and sections we build up (left) loops (L,∗) contained in the group G where G=PGL2(q) or G=PSL2(q), q>3 odd. Since such groups can be embedded in PG(3,q), the subgroups we considered for the construction of (L,∗) are geometrically relevant (lines or planes of PG(3,q)) or algebraically relevant (maximal subgroups). [ABSTRACT FROM AUTHOR]

We determine the multiplicative loops of locally compact connected 4-dimensional quasifields Q having the field of complex numbers as their kernel. In particular, we turn our attention to multiplicative loops which have either a normal subloop of dimension one or which contain a subgroup isomorphic to $$Spin_3({\mathbb {R}})$$ . Although the 4-dimensional semifields Q are known, their multiplicative loops have interesting Lie groups generated by left or right translations. We determine explicitly the quasifields Q which coordinatize locally compact translation planes of dimension 8 admitting an at least 16-dimensional Lie group as automorphism group. [ABSTRACT FROM AUTHOR]

Let $$L=M(G,2)$$ be a Moufang loop, F[ L] be its loop algebra over a field F and J( F[ L]) be the Jacobson radical of F[ L]. In this article, we obtain the unit loop of F[ L] / J( F[ L]) and the structure of $$1+J(F[L])$$ , where $$L=M(G,2)$$ is obtained from the generalized dihedral group $$G=Dih(C_p^2)$$ , p an odd prime and F is a finite field of characteristic 2 containing a primitive pth root of unity. [ABSTRACT FROM AUTHOR]

It is known that all Moufang loops of order $$p^4$$ are associative if p is a prime greater than 3. Also, nonassociative Moufang loops of order $$p^5$$ (for all primes p) and $$pq^3$$ (for distinct odd primes p and q, with the necessary and sufficient condition $$q\equiv 1({\text{ mod }}\ p)$$ ) have been proved to exist. Consider a Moufang loop L of order $$p^{\alpha }q^{\beta }$$ where p and q are odd primes with $$p3$$ , $$\alpha \le 4$$ and $$\beta \le 3$$ . [ABSTRACT FROM AUTHOR]

In this article, we describe a key exchange protocol based on right transversals. We also describe it for general extension associated to right loops. [ABSTRACT FROM PUBLISHER]

Möbius addition is defined on the complex open unit disk by and Möbius's exponential equation takes the form $$L(a\oplus _M b) = L(a)L(b)$$ , where L is a complex-valued function defined on the complex unit disk. In the present article, we indicate how Möbius's exponential equation is connected to Cauchy's exponential equation. Möbius's exponential equation arises when one determines the irreducible linear representations of the unit disk equipped with Möbius addition, considered as a nonassociative group-like structure. This suggests studying Schur's lemma in a more general setting. [ABSTRACT FROM AUTHOR]

Automorphic loops, orA-loops, are loops in which all inner mappings are automorphisms. We investigatedA-loops arising from a Lie algebra and describe their automorphism group. Also, we identify and describe their inner mapping group. [ABSTRACT FROM AUTHOR]

We define a variety of loops called semiautomorphic, inverse property loops that generalize Moufang and Steiner loops. We first show an equivalence between a previously studied variety of loops. Next we extend several known results for Moufang and Steiner loops. That is, the commutant is a subloop and ifais in the commutant, thena2is a Moufang element,a3is ac-element anda6is in the center. Finally, we give two constructions for semiautomorphic inverse property loops based on Chein’s and de Barros and Juriaans’ doubling constructions. [ABSTRACT FROM AUTHOR]

The first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so-called pseudo-orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo-orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo-orbits. An upper bound is given on the size of a pseudo-orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15. [ABSTRACT FROM AUTHOR]

We show that there does not exist any connected topological proper loop homeomorphic to a quasi-simple Lie group and having a compact Lie group as the group topologically generated by its left translations. Moreover, any connected topological loop homeomorphic to the 7-sphere and having a compact Lie group as the group of its left translations is classical. We give a particular simple general construction for proper loops such that the compact group of their left translations is direct product of at least 3 factors. [ABSTRACT FROM AUTHOR]

A representation of an object in a category is an abelian group in the corresponding comma category. In this paper, we derive the formulas describing linear representations of objects in the category of formal loops and formal loop homomorphisms and apply them to obtain a new approach to the representation theory of formal Moufang loops and Malcev algebras based on Moufang elements. Certain ‘non-associative Moufang symmetry’ of groups is revealed. [ABSTRACT FROM AUTHOR]

The equational variety of quasigroups is defined by six identities, called Birkhoff's identities. It is known, that only four of them suffice to define the variety; actually, there are nine different combinations of four Birkhoff's identities defining quasigroups, other four combinations define larger varieties and it was open whether the remaining two cases define quasigroups or larger classes. We solve the question here constructing examples of algebras that are not quasigroups and satisfy the open cases of Birkhoff's identities. [ABSTRACT FROM AUTHOR]