Commutative Loops Research Articles

Commutative quasigroups and semifields from planar functions

Finite commutative semifields of odd characteristic correspond to Dembowski-Ostrom polynomials. A new proof of this fact is the main result of this paper. The paper also discusses strong isotopy of commutative semifields and shows that the limit on the number of strong isotopy classes can be obtained from a general theorem on commutative loops. By this theorem for each commutative loop Q the commutative isotopes form at most | N μ : ( N μ ) 2 N λ | classes with respect to the strong isotopy, where N μ and N λ are the middle and the left nucleus of Q. Loops Q 1 and Q 2 are said to be strongly isotopic if there exists an isotopism Q 1 → Q 2 of the form ( α , α , γ ) .

Automorphic loops and metabelian groups

Given a uniquely 2-divisible group $G$, we study a commutative loop $(G,\circ)$ which arises as a result of a construction in ``Engelsche elemente noetherscher gruppen'' (1957) by R. Baer. We investigate some general properties and applications of ``$\circ$'' and determine a necessary and sufficient condition on $G$ in order for $(G, \circ)$ to be Moufang. In ``A class of loops categorically isomorphic to Bruck loops of odd order'' (2014) by M. Greer, it is conjectured that $G$ is metabelian if and only if $(G, \circ)$ is an automorphic loop. We answer a portion of this conjecture in the affirmative: in particular, we show that if $G$ is a split metabelian group of odd order, then $(G, \circ)$ is automorphic.

Basarab loop and the generators of its total multiplication group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop

Basarab loop and the generators of its total multiplication group

A loop (Q; ·) is called a Basarab loop if the identities: (x · yxρ)(xz) = x · yz and (yx) · (xλz · x) = yz · x hold. It was shown that the left, right and middle nuclei of the Basarab loop coincide, and the nucleus of a Basarab loop is the set of elements x whose middle inner mapping Tx are automorphisms. The generators of the inner mapping group of a Basarab loop were refined in terms of one of the generators of the total inner mapping group of a Basarab loop. Necessary and su_cient condition(s) in terms of the inner mapping group (associators) for a loop to be a Basarab loop were established. It was discovered that in a Basarab loop: the mapping x ↦ Tx is an endomorphism if and only if the left (right) inner mapping is a left (right) regular mapping. It was established that a Basarab loop is a left and right automorphic loop and that the left and right inner mappings belong to its middle inner mapping group. A Basarab loop was shown to be an automorphic loop (A-loop) if and only if it is a middle automorphic loop (middle A-loop). Some interesting relations involving the generators of the total multiplication group and total inner mapping group of a Basarab loop were derived, and based on these, the generators of the total inner mapping group of a Basarab loop were finetuned. A Basarab loop was shown to be a totally automorphic loop (TA-loop) if and only if it is a commutative and exible loop. These aforementioned results were used to give a partial answer to a 2013 question and an ostensible solution to a 2015 problem in the case of Basarab loop.

A correspondence between commutative rings and Jordan loops

We show that there is a one-to-one correspondence (up to isomorphism) between commutative rings with unity and metabelian commutative loops belonging to a particular finitely axiomatizable class. Based on this correspondence, it is proved that the sets of identically valid formulas and of finitely refutable formulas of a class of finite nonassociative commutative loops (and of many of its other subclasses) are recursively inseparable. It is also stated that nonassociative commutative free automorphic loops of any nilpotency class have an undecidable elementary theory.

On torsion-free nilpotent loops

Abstract We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotence of the free loop proved by Higman, it is established, essentially, by the same method.

On loop extensions satisfying one single identity and cohomology of loops

ABSTRACTIn this paper are defined cohomology-like groups that classify loop extensions satisfying a given identity in three variables for association identities and in two variables for the case of commutativity. It is considered a large amount of identities. These groups generalize those defined in Nishigori [3] and of Kenneth and Leedham-Green [2]. It is computed the number of metacyclic extensions for trivial action of the quotient on the kernel in one particular case for left Bol loops and in general for commutative loops.

On non-associative generalizations of MV-algebras and lattice-ordered commutative loops

The aim of the paper is to describe the connections between lattice-ordered commutative loops and certain “basic algebras” which are a non-associative generalization of MV-algebras and are related to commutative semicopulas. This extends the well-known equivalence between lattice-ordered Abelian groups with strong order-unit and MV-algebras.

On finite commutative loops which are centrally nilpotent

Let $Q$ be a finite commutative loop and let the inner mapping group $I(Q) \cong C_{p^n} \times C_{p^n}$, where $p$ is an odd prime number and $n \geq 1$. We show that $Q$ is centrally nilpotent of class two.

A Class of Loops Categorically Isomorphic to Bruck Loops of Odd Order

We define a new variety of loops, Γ-loops. After showing Γ-loops are power-associative, our main goal is showing a categorical isomorphism between Bruck loops of odd order and Γ-loops of odd order. Once this has been established, we can use the well known structure of Bruck loops of odd order to derive the Odd Order, Lagrange and Cauchy Theorems for Γ-loops of odd order, as well as the nontriviality of the center of finite Γ-p-loops (p odd). Finally, we answer a question posed by Jedlička, Kinyon and Vojtěchovský about the existence of Hall π-subloops and Sylow p-subloops in commutative automorphic loops.

A Construction of Right Alternative Loop

Right alternative loops are loops satisfying \((xy)y=x(yy)\). We construct an infinite family of non-associative non commutative right alternative loops whose smallest member is of order \(8\), without any usage of computers.

Automorphisms of commutative Moufang loops satisfying the minimality condition

We consider commutative Moufang loops Q with multiplicative group M satisfying the minimality condition for its subloops. Such loops, as well as the class of such loops, are characterized by various subgroups of automorphism groups Aut Q and Aut M. We study the structure of the groups Aut Q and Aut M and prove that these groups have matrix representations.

Constructions of Commutative Automorphic Loops

A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate the classification of commutative A-loops of small orders and also of order p 3, where p is a prime.

Groupoids with the identity defining the commutative Moufang loops

It is known that in the class of loops the identity xy · xz = x 2 · yz defines the commutative Moufang loops. In the article, we consider groupoids with this identity and establish properties of some classes of such groupoids.

Algebraic structure in a family of Nim-like arrays

We study aspects of the algebraic structure shared by a certain family of recursively generated arrays related to the operation of Nim-addition. We first observe that each individual array represents a countably infinite, commutative loop (in the sense of quasigroups). We then prove that each loop in the family is monogenic (generated by a single element in a non-associative fashion), and use this to determine all loop homomorphisms between members of the family.

Primary Decompositions in Varieties of Commutative Diassociative Loops

The decomposition theorem for torsion abelian groups holds analogously for torsion commutative diassociative loops. With this theorem in mind, we investigate commutative diassociative loops satisfying the additional condition (trivially satisfied in the abelian group case) that all nth powers are central, for a fixed n. For n = 2, we get precisely commutative C loops. For n = 3, a prominent variety is that of commutative Moufang loops. Many analogies between commutative C and Moufang loops have been noted in the literature, often obtained by interchanging the role of the primes 2 and 3. We show that the correct encompassing variety for these two classes of loops is the variety of commutative RIF loops. In particular, when Q is a commutative RIF loop: all squares in Q are Moufang elements, all cubes are C elements, Moufang elements of Q form a normal subloop M 0(Q) such that Q/M 0(Q) is a C loop of exponent 2 (a Steiner loop), C elements of L form a normal subloop C 0(Q) such that Q/C 0(Q) is a Moufang loop of exponent 3. Since squares (resp., cubes) are central in commutative C (resp., Moufang) loops, it follows that Q modulo its center is of exponent 6. Returning to the decomposition theorem, we find that every torsion, commutative RIF loop is a direct product of a C 2-loop, a Moufang 3-loop, and an abelian group with each element of order prime to 6. We also discuss the definition of Moufang elements and the quasigroups associated with commutative RIF loops.

Admissible Orders of Jordan Loops

AbstractA commutative loop is Jordan if it satisfies the identity x2(yx) = (x2y)x. Using an amalgam construction and its generalizations, we prove that a nonassociative Jordan loop of order n exists if and only if n≧ 6 and n≠ 9. We also consider whether powers of elements in Jordan loops are well‐defined, and we construct an infinite family of finite simple nonassociative Jordan loops. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 103–118, 2009

Shortest single axioms for commutative moufang loops of exponent 3

In this note, we attempt to find all shortest single product axioms for commutative Moufang loops of exponent 3. These investigations were aided by the automated theorem-prover Prover9 and the model-generator Mace4.

Structure of finitely generated commutative alternative algebras and special Moufang loops

It is proved that the commutator ideal of the multiplication algebra of a free commutative alternative algebra of rank n is nilpotent of index n − 1. As a corollary to this fact, the Bruck theorem for special commutative Moufang loops is derived.

Commutative free loop space models at large primes

Keywords: free loop spaces ; algebraic models ; Adams-Hilton models ; Anick spaces Reference GR-HE-ARTICLE-2003-002doi:10.1007/s00209-003-0206-yView record in Web of Science Record created on 2008-11-14, modified on 2017-05-12