Related papers: C-loops: An introduction
C-loops are loops satisfying the identity $x(y\cdot yz) = (xy\cdot y)z$. We develop the theory of extensions of C-loops, and characterize all nuclear extensions provided the nucleus is an abelian group. C-loops with central squares have…
Isotopes of C-loops with unique non-identity squares are shown to be both C-loops and A-loops. The relationship between C-loops and Steiner loops is further studied. Central loops with the weak and cross inverse properties are also…
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…
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…
We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If $Q$ is a PACC-loop with nucleus $N$, then $Q/N$ is an abelian group of exponent 12; if in addition $Q$ is finite, then $|Q|$ is divisible by 16 or by…
A loop $X$ is said to satisfy Moufang's theorem if for every $x,y,z\in X$ such that $x(yz)=(xy)z$ the subloop generated by $x$, $y$, $z$ is a group. We prove that the variety $V$ of Steiner loops satisfying the identity $(xz)(((xy)z)(yz)) =…
An algebraic process for the construction of an autotopism for a non-Steiner C-loop is described and this is demonstrated with an example using a known finite C-loop. In every C-loop, two of its parastrophes are not equivalent(equal) it, if…
It is proved that the following conditions are equivalent for an infinite non-associative commutative Moufang loop $Q$: 1) $Q$ satisfies the minimum condition for subloops; 2) if the loop $Q$ contains a centrally solvable subloop of class…
We study a variety of loops, RIF, which arise naturally from considering inner mapping groups, and a somewhat larger variety, ARIF. All Steiner and Moufang loops are RIF, and all flexible C-loops are ARIF. We show that all ARIF loops are…
The structure of the commutative Moufang loops (CML) with minimum condition for subloops is examined. In particular it is proved that such a CML $Q$ is a finite extension of a direct product of a finite number of the quasicyclic groups,…
LC-loops, RC-loops and C-loops are collectively called central loops. It is shown that an LC(RC)-loop is a left(right) universal loop. But an LC(RC)-loop is a universal loop if and only if it is a right(left) universal loop. It is observed…
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…
Andrew Rajah posed at the Loops'11 Conference in Trest, Czech Republic, the following conjecture: Is every variety of loops that satisfies Moufang's theorem contained in the variety of Moufang loops? This paper is motivated by that problem.…
Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a…
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…
Although little can be gleaned about a loop with the property that its squares are, say, left nuclear ($xx\cdot yz = (xx\cdot y)z$), if its squares are also, say, middle nuclear ($(x\cdot yy)z = x(yy\cdot z)$), then the loop exhibits more…
A loop whose inner mappings are automorphisms is an \emph{automorphic loop} (or \emph{A-loop}). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain…
We use groups with triality to construct a series of nonassociative Moufang loops. Certain members of this series contain an abelian normal subloop with the corresponding quotient being a cyclic group. In particular, we give a new series of…
A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit…
We construct two infinite series of Moufang loops of exponent $3$ whose commutative center (i.e. the set of elements that commute with all elements of the loop) is not a normal subloop. In particular, we obtain examples of such loops of…