Search

Let V be a cubic surface defined by the equation T 0 3 + T 1 3 + T 2 3 + θ T 3 3 = 0 over a quadratic extension of 3-adic numbers k = Q 3 (θ) , where θ 3 = 1 . We show that a relation on a set of geometric k-points on V modulo (1 - θ) 3 (in a ring of integers of k) defines an admissible relation and a commutative Moufang loop associated with classes of this admissible equivalence is non-associative. This answers a problem that was formulated by Yu. I. Manin more than 50 years ago about existence of a cubic surface with a non-associative Moufang loop of point classes. [ABSTRACT FROM AUTHOR]

Let O(k) be the octonion Cayley-Dickson algebra over a commutative associative ring κ with 1. Let G(k) be the Moufang loop of invertible elements of O(k). Let H be a class of groups such that a group G is a member of H if and only if G satisfies the following three conditions: (a) G is not class-2 nilpotent. (b) G has a proper class-2 nilpotent subgroup. (c) G is not isomorphic to any subgroup of the group GL2(F) for any field F. The theorem proved in the paper states that if κ is an integral domain with 1+1 6= 0, then G(k) does not contain any subloop isomorphic to a group of class H, while if κ is an integral domain such that 1+1 = 0, then G(k) contains no subloop isomorphic to a class-2 nilpotent group at all. [ABSTRACT FROM AUTHOR]

Let K be a field, O(K) the split Cayley–Dickson (octonion) algebra over K, and G(K) the Moufang loop formed by all elements of O(K) having norm other than 0. A transvection is an element of G(K) having the form 1 + a , where a = 0 and a 2 = 0. In the paper, subloops of G(K), containing a subloop which consists only of transvections, are studied. [ABSTRACT FROM AUTHOR]

We shall show that there exist sofic groups which are not locally embeddable into finite Moufang loops. These groups serve as counterexamples to a problem and two conjectures formulated in the paper by M.Vodička, P. Zlatoš (2019). [ABSTRACT FROM AUTHOR]

A loop (Q, ∙) is called a FRUTE loop if it obeys the identity (x ∙ xy)z = (y ∙ zx)x. Interestingly, a FRUTE loop is a Moufang loop, but not neccessarily an extra loop or a group (and vice versa). In this paper, some new algebraic properties of a FRUTE loop are established. It is shown that a FRUTE loop is centrum square, a middle A-Ioop and obeys the law x-1yx = xyx-1. It is proved that in a FRUTE loop, the following loops are equivalent: central square loop, group, extra loop, LC-Ioop, RC-Ioop, C-loop, LNS-loop, RNS-loop, MNS-loop, left Cheban loop, right Cheban loop, Cheban loop, LCC-loop, RCC-Ioop and CC-Ioop. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring with 1, M an ideal of A, E a power associative algebra over A having a basis and a unit element e. In the paper, the torsion in the multiplicatively closed subset e + ME of E has been studied when A is an integral domain of characteristic 0 with a theory of divisors. The main theorem of the paper generalizes a result concerning the torsion in the congruence subgroup of the general linear group over A. [ABSTRACT FROM AUTHOR]

Let H = (V, F) be a simple hypergraph without loops. H is called linear if |f ∩g| ≤ 1 for any f, g ∈ F with f 6= g. The 2-section of H, denoted by [H]2, is a graph with V ([H]2) = V and for any u, v ∈ V ([H]2), uv ∈ E([H]2) if and only if there is f ∈ F such that u, v ∈ f. The treewidth of a graph is an important invariant in structural and algorithmic graph theory. In this paper, we consider the treewidth of the 2-section of a linear hypergraph. We will use the minimum degree, maximum degree, anti-rank and average rank of a linear hypergraph to determine the upper and lower bounds of the treewidth of its 2-section. Since for any graph G, there is a linear hypergraph H such that [H]2 ∼= G, we provide a method to estimate the bound of treewidth of graph by the parameters of the hypergraph. [ABSTRACT FROM AUTHOR]

In this note we describe the structure of finite-dimensional Malcev algebras over the field of real numbers R , which are nilpotent modulo its Lie center. It is proved that the corresponding analytic global Moufang loops are nilpotent modulo their nucleus. [ABSTRACT FROM AUTHOR]

We construct a non-Moufang variety of Steiner loops H such that every loop from H satisfies Moufang's theorem. This example solves Rajah's problem on the existence of such varieties. [ABSTRACT FROM AUTHOR]

This article studies subloops of the Moufang loop of invertible elements in the split Cayley–Dickson algebra over a field K containing a nonassociative subloop formed by elements of the split Cayley–Dickson algebra over a field k such that K is an algebraic extension of k. [ABSTRACT FROM AUTHOR]

A loop (Q, -, \, /) is called a middle Bol loop (MBL) if it obeys the identity x(yz\x)=(x/z)(y\x). To every MBL corresponds a right Bol loop (RBL) and a left Bol loop (LBL). In this paper, some new algebraic properties of a middle Bol loop are established in a different style. Some new methods of constructing a MBL by using a nonabelian group, the holomorph of a right Bol loop and a ring are described. Some equivalent necessary and sufficient conditions for a right (left) Bol loop to be a middle Bol loop are established. A RBL (MBL, LBL, MBL) is shown to be a MBL (RBL, MBL, LBL) if and only if it is a Moufang loop. [ABSTRACT FROM AUTHOR]

The intersection graph ΓSI (G) of a group G with identity element e is the graph whose vertex set is the set V (ΓSI (G)) = G - e and two distinct vertices x and y are adjacent in ΓSI (G) if and only if |〈x〉∩〈y〉| > 1, where hxi is the cyclic subgroup of G generated by x. In this paper, at first we obtain some results for this graph for any Moufang loop. More specially we observe non-isomorphic finite Moufang loops may have isomorphic intersection graphs. [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]

A loop L is strongly admissible if there exists a bijection ϕ : L → L for which the mappings g ↦ g ϕ (g) and g ↦ g (g ϕ (g)) are both bijections. We derive some results on the strong admissibility of finite nonassociative Moufang loops, constructed from groups using the Chein constructions. [ABSTRACT FROM AUTHOR]

Left weak inverse property, power associative conjugacy closed (LWPC) quasigroups are quasigroups that satisfy the identity (xy · x) . xz = x((yx·x)z). Right weak inverse property, power associative conjugacy closed (RWPC) quasigroups satisfy the mirror identity zx · (x · yx) = (z(x · xy))x. It is shown that LWPC quasigroups are flexible, satisfy the inverse property and are alternative. It is shown that an LWPC quasigroup is a loop. [ABSTRACT FROM AUTHOR]

In this note, we introduce the concept of an almost left automorphic Moufang loop and study the properties of tangent algebras of smooth loops of this class. [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]

We show that any simple group of Morley rank 5 is a bad group all of whose proper definable connected subgroups are nilpotent of rank at most 2. The main result is then used to catalog the nonsoluble connected groups of Morley rank 5. [ABSTRACT FROM AUTHOR]

Properties of left-transitive quasigroups (including their autotopisms and pseudoautomorphisms), their connections with some quasigroup classes are established. Left-transitive right GA-quasigroups are described. [ABSTRACT FROM AUTHOR]

The non-commuting graph associated to a non-abelian group , , is a graph with vertex set where distinct non-central elements and of are joined by an edge if and only if . The non-commuting graph of a non-abelian finite group has received some attention in existing literature. Recently, many authors have studied the non-commuting graph associated to a non-abelian group. In particular, the authors put forward the following conjectures: Conjecture 1. Let and be two non-abelian finite groups such that . Then . Conjecture 2 (AAM's Conjecture). Let be a finite non-abelian simple group and be a group such that . Then . Some authors have proved the first conjecture for some classes of groups (specially for all finite simple groups and non-abelian nilpotent groups with irregular isomorphic non-commuting graphs) but in [Moghaddamfar, About noncommuting graphs, Sib. Math. J. 47(5) (2006) 911-914], Moghaddamfar has shown that it is not true in general with some counterexamples to this conjecture. On the other hand, Solomon and Woldar proved the second conjecture, in [R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graph, J. Group Theory 16 (2013) 793-824]. In this paper, we will define the same concept for a finite non-commutative Moufang loop and try to characterize some finite non-commutative Moufang loops with their non-commuting graph. Particularly, we obtain examples of finite non-associative Moufang loops and finite associative Moufang loops (groups) of the same order which have isomorphic non-commuting graphs. Also, we will obtain some results related to the non-commuting graph of a finite non-commutative Moufang loop. Finally, we give a conjecture stating that the above result is true for all finite simple Moufang loops. [ABSTRACT FROM AUTHOR]

In this work, we construct free infinitely generated Moufang loop in the variety generated by code loops and find the minimal set of identities that define this variety. We apply this construction to the study of code loops. [ABSTRACT FROM AUTHOR]

It is known that semigroups are Ramsey algebras. This paper is an attempt to understand the role associativity plays in a binary system being a Ramsey algebra. Specifically, we show that the nonassociative Moufang loop of octonions is not a Ramsey algebra. [ABSTRACT FROM AUTHOR]

In this paper, we extend the theory of special local Moufang sets. We construct a local Moufang set from every local Jordan pair, and we show that every local Moufang set satisfying certain (natural) conditions gives rise to a local Jordan pair. We also explore the connections between these two constructions. [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]

Here cyclic extensions, not necessarily split, of simple groups are looked at. It is shown that if N is a finite simple group that is either an abelian group, an alternating group, a Suzuki group, a projective special linear group or a sporadic simple group and G = (N; u) is a cyclic extension of N resulting in a Moufang loop, then (N; u²) is a group. Moreover, if G is nonassociative, then G is a generalization of the Chein loop where u inverts all of the elements of N. [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]

We enumerate three classes of non-medial quasigroups of order 243 = 3 5 up to isomorphism. There are 17 004 non-medial trimedial quasigroups of order 243 (extending the work of Kepka, Bénéteau and Lacaze), 92 non-medial distributive quasigroups of order 243 (extending the work of Kepka and Němec), and 6 non-medial distributive Mendelsohn quasigroups of order 243 (extending the work of Donovan, Griggs, McCourt, Opršal and Stanovský). The enumeration technique is based on affine representations over commutative Moufang loops, on properties of automorphism groups of commutative Moufang loops, and on computer calculations with the LOOPS package in GAP . [ABSTRACT FROM AUTHOR]

We prove a purely topological characterization of the Moufang property for disconnected compact polygons in terms of convergence groups. As a consequence, we recover the fact that a locally finite thick affine building of rank 3 is a Bruhat-Tits building if and only if its automorphism group is strongly transitive. We also study automorphism groups of general compact polygons without any homogeneity assumption. A compactness criterion for sets of automorphisms is established, generalizing the theorem by Burns and Spatzier that the full automorphism group, endowed with the compact-open topology, is a locally compact group. [ABSTRACT FROM AUTHOR]

A loop is called a FRUTE loop if it obeys the identity . Interestingly, a FRUTE loop is a Moufang loop but not necessarily an extra loop or a group (and vice versa). In this paper, algebraic properties of the left (right) regular representation set of a FRUTE loop are deduced. A FRUTE loop is shown to be universal and an -loop for all . A Moufang loop is shown to be a FRUTE loop if and only if it is nuclear cube if and only if it is an -loop. It is established that: the smallest, associative, non-commutative FRUTE loop is of order (the quaternion group ); for any , there exists at least a non-commutative group of order that is a FRUTE loop; there exists -groups of orders that are non-commutative FRUTE loops; there are no non-commutative groups that are FRUTE loops of the following range of orders ; there are two non-associative FRUTE loops of order up to isomorphism and there are six non-isomorphic, non-associative FRUTE loops of order . It is noted that there exists a non-associative and non-commutative FRUTE loop of order . The study is concluded with some questions, conjectures and problem. [ABSTRACT FROM AUTHOR]

A Moufang loop M of order 3 is constructed, together with a pair a, b of elements of M, such that the set of all elements of M associating with a and b is not a subloop. This also gives an example of a nonassociative Moufang loop with a generating set in which every three elements have trivial associator. [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]

Glauberman and Wright in [G. G. Glauberman and C. R. B. Wright, Nilpotence of finite Moufang 2-loops, J. Algebra 8 (1968) 415-417] proved that a nilpotent Moufang loop is the direct product of -loops for some primes , consequently the elements of coprime order commute in a nilpotent Moufang loop. In this paper, we prove that in Moufang loops of odd order this condition is equivalent to the central nilpotence. [ABSTRACT FROM AUTHOR]

In the mid 1970s, Michel Racine classified the maximal subalgebras of an octonian algebra. In this paper, we classify the maximal associative subalgebras. It turns out that there are four, up to isomorphism, all of dimension . In final sections, we apply our findings to investigate the groups that sit inside the Moufang loop of invertible elements of the split octonians and also to show that a well-known inequality of Jørgensen holds in a new context. [ABSTRACT FROM AUTHOR]

Quantum quasigroups are algebraic structures providing a general self-dual framework for the nonassociative extension of Hopf algebra techniques. They also have one-sided analogues, which are not self-dual. The paper presents a survey of recent work on these structures, showing how they furnish various solutions to the quantum Yang--Baxter equation. [ABSTRACT FROM AUTHOR]

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop Q of odd order is solvable and contains an element of order p for every prime p dividing |Q|, and that |S| divides |Q| for every subloop S of Q. There are no finite simple nonassociative commutative automorphic loops, and there are no finite simple nonassociative automorphic loops of order less than 2500. We show that if Q is a finite simple nonassociative automorphic loop, then the socle of the multiplication group of Q is not regular. The existence of a finite simple nonassociative automorphic loop remains open. Let p be an odd prime. Automorphic loops of order p or p² are groups, but there exist nonassociative automorphic loops of order p³, some with trivial nucleus (center) and of exponent p. We construct nonassociative "dihedral" automorphic loops of order 2n for every n > 2, and show that there are precisely p-2 nonassociative automorphic loops of order 2p, all of them dihedral. [ABSTRACT FROM AUTHOR]

We prove that the unipotent horocyclic group of a Moufang twin tree of prime order is nilpotent of class at most 2. [ABSTRACT FROM AUTHOR]

We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general, this is not true for finite left automorphic Moufang loops and for finite automorphic loops. [ABSTRACT FROM AUTHOR]

The commutativity degree, Pr(G), of a finite group G (i.e. the probability that two (ran-domly chosen) elements of G commute with respect to its operation)) has been studied well by many authors. It is well-known that the best upper bound for Pr(G) is 5/8 for a finite non-abelian group G. In this paper, we will define the same concept for a finite non-abelian Moufang loop M and try to give a best upper bound for Pr(M). We will prove that for a well-known class of finite Moufang loops, named Chein loops, and its modifications, this best upper bound is 23/32. So, our conjecture is that for any finite Moufang loop M, Pr(M) ≤ 23/32. Also, we will obtain some results related to the Pr(M) and ask the similar questions raised and answered in group theory about the relations between the structure of a finite group and its commutativity degree in finite Moufang loops. [ABSTRACT FROM AUTHOR]

We introduce local Moufang sets as a generalization of Moufang sets. We present a method to construct local Moufang sets from only one root group and one permutation. We use this to describe PSL2 over a local ring as a local Moufang set, and give necessary and sufficient conditions for a local Moufang set to be of this form. [ABSTRACT FROM AUTHOR]

We consider the relationship between Mal'tsev bialgebras and Lie bialgebras with triality, and also between symplectic Mal'tsev algebras and symplectic Lie algebras with triality. The given relations generalize a connection between Mal'tsev algebras and Lie algebras with triality, revealed by P. O. Mikheev [15], and a connection between Mal'tsev coalgebras and Lie coalgebras with triality, explored by M. E. Goncharov and V. N. Zhelyabin [17]. [ABSTRACT FROM AUTHOR]

Using groups with triality, we obtain some general multiplication formulas in Moufang loops, construct Moufang extensions of abelian groups, and describe the structure of minimal extensions for finite simple Moufang loops over abelian groups. [ABSTRACT FROM AUTHOR]

A 3- net of order n is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around 3-nets (embedded) in a projective plane $$\mathrm{PG}(2,\mathbb {K})$$ , defined over a field $$\mathbb {K}$$ of characteristic p, arose from algebraic geometry; see Falk and Yuzvinsky (Compos Math 143:1069-1088, 2007), Miguel and Buzunáriz (Graphs Comb 25:469-488, 2009), Pereira and Yuzvinsky (Adv Math 219:672-688, 2008), Yuzvinsky (140:1614-1624, 2004), and Yuzvinsky (137:1641-1648, 2009). It is not difficult to find 3-nets in $$\mathrm{PG}(2,\mathbb {K})$$ as far as $$0n$$ . Under this condition, the known families are characterized as the only 3-nets in $$\mathrm{PG}(2,\mathbb {K})$$ which can be coordinatized by a group; see Korchmáros et al. (J Algebr Comb 39:939-966, 2014). In this paper we deal with 3-nets in $$PG(2,\mathbb {K})$$ which can be coordinatized by a diassociative loop G but not by a group. We prove two structural theorems on G. As a corollary, if G is commutative then every non-trivial element of G has the same order, and G has exponent 2 or 3 where the exponent of a finite diassociative loop is the maximum of the orders of its elements. We also discuss the existence problem for such 3-nets. [ABSTRACT FROM AUTHOR]

We introduce a class of non-Moufang loops satisfying Moufang's theorem. [ABSTRACT FROM AUTHOR]

We prove that if the squaring map in the factor loop of a Moufang loop Q over its nucleus is surjective, then every half-isomorphism of Q onto a Moufang loop is either an isomorphism or an anti-isomorphism. This generalizes all earlier results in this vein. [ABSTRACT FROM AUTHOR]

Let F[L] be a loop algebra of a loop L over a Field F. In this paper, we obtain the unit loop of the loop algebra F[L]; where L is the smallest Moufang loop M(S3, 2) and F is a nite field of characteristic different from 3. [ABSTRACT FROM AUTHOR]