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On Moufang A-loops

Jon D. Phillips (2000)

Commentationes Mathematicae Universitatis Carolinae

In a series of papers from the 1940’s and 1950’s, R.H. Bruck and L.J. Paige developed a provocative line of research detailing the similarities between two important classes of loops: the diassociative A-loops and the Moufang loops ([1]). Though they did not publish any classification theorems, in 1958, Bruck’s colleague, J.M. Osborn, managed to show that diassociative, commutative A-loops are Moufang ([5]). In [2] we relaunched this now over 50 year old program by examining conditions under which...

On multiplication groups of left conjugacy closed loops

Aleš Drápal (2004)

Commentationes Mathematicae Universitatis Carolinae

A loop Q is said to be left conjugacy closed (LCC) if the set { L x ; x ∈ Q } is closed under conjugation. Let Q be such a loop, let ℒ and ℛ be the left and right multiplication groups of Q , respectively, and let Inn Q be its inner mapping group. Then there exists a homomorphism ℒ → Inn Q determined by L x ↦ R x - 1 L x , and the orbits of [ ℒ , ℛ ] coincide with the cosets of A ( Q ) , the associator subloop of Q . All LCC loops of prime order are abelian groups.

On multiplication groups of relatively free quasigroups isotopic to Abelian groups

Aleš Drápal (2005)

Czechoslovak Mathematical Journal

If Q is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group M l t Q is a Frobenius group. Conversely, if M l t Q is a Frobenius group, Q a quasigroup, then Q has to be isotopic to an Abelian group. If Q is, in addition, finite, then it must be a central quasigroup (a T -quasigroup).

On nondistributive Steiner quasigroups

A. Marczak (1997)

Colloquium Mathematicae

A well known result of R. Dedekind states that a lattice is nonmodular if and only if it has a sublattice isomorphic to N 5 . Similarly a lattice is nondistributive if and only if it has a sublattice isomorphic to N 5 or M 3 (see [11]). Recently a few results in this spirit were obtained involving the number of polynomials of an algebra (see e.g. [1], [3], [5], [6]). In this paper we prove that a nondistributive Steiner quasigroup (G,·) has at least 21 essentially ternary polynomials (which improves the...

On permutable N-groups

W. A. Dudek, Z. Stojakovic (1988)

Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry

On quasigroups with the left loop property

A. Nourou Issa (2000)

Commentationes Mathematicae Universitatis Carolinae

Some properties of quasigroups with the left loop property are investigated. In loops we point out that the left loop property is closely related to the left Bol identity and the particular case of homogeneous loops is considered.

On quasivarieties of nilpotent Moufang loops. II

Vasile I. Ursu (2012)

Commentationes Mathematicae Universitatis Carolinae

In this part of the paper we study the quasiidentities of the nilpotent Moufang loops. In particular, we solve the problem of finite basis for quasiidentities in the finitely generated nilpotent Moufang loop.

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