arXiv subjects
Publications and source records attributed to J. D. Phillips.
In math.GR/0510298, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the equational class of (pointed) F-quasigroups and the equational class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.
We describe all constructions for loops of Bol-Moufang type analogous to the Chein construction $M(G,*,g_0)$ for Moufang loops.
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 very transparent extensions; they can be built from small blocks arising from the underlying Steiner triple system. Using these extensions, we decide for which abelian groups $K$ and Steiner loops $Q$ there is a nonflexible C-loop $C$ with center $K$ such that $C/K$ is isomorphic to $Q$. We discuss possible orders of associators in C-loops. Finally, we show that the loops of signed basis elements in the standard real Cayley-Dickson algebras are C-loops.
We solve two problems posed by Krapež by finding a basis of seven independent axioms for the variety of rectangular loops. Six of these axioms form a basis for the variety of rectangular quasigroups. The proofs of the lemmas showing that the six axioms are sufficient are based on proofs generated by the automated reasoning program OTTER, while most of the models verifying the independence of the axioms were generated by the finite model builder Mace4.
In the spirit of Glauberman's fundamental work in B-loops and Moufang loops, we prove Cauchy and strong Lagrange theorems for Bol loops of odd order. We also establish necessary conditions for the existence of a simple Bol loop of odd order, conditions which should be useful in the development of a Feit-Thompson theorem for Bol loops. Bol loops are closely related to Aschbacher's twisted subgroups, and we survey the latter in some detail, especially with regard to the so-called Aschbacher radical.
We give new equations that axiomatize the variety of trimedial quasigroups. We also improve a standard characterization by showing that right semimedial, left F-quasigroups are trimedial.
A loop $(X,\circ)$ is said to be a Bruck loop if it satisfies the (right) Bol identity $((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x)$ and the automorphic inverse property $(x\circ y)^{-1}=x^{-1}\circ y^{-1}$. If $X$ is a finite Bruck loop and $G$ is the group generated by all right translations $R(x): y\mapsto y\circ x$, then we show that $X$ and $G$ are central products $X = O^{2'}(X) * O(X)$ and $G = O^{2'}(G) * O(G)$, where $O^{2'}(X)$ ($O^{2'}(G)$) is the subloop (subgroup) generated by all 2-elements, and $O(X)$ ($O(G)$) is the largest normal subloop (subgroup) of odd order. In particular, if $X$ is solvable, then these central products are direct products. We also give a set of necessary conditions that must hold for a finite Bruck loop $X$ to be nonsolvable but have each proper section solvable; in particular, $X$ must be simple and consist of 2-elements, while the quotient of $G$ by its largest normal 2-subgroup must be isomorphic to $PGL_2(q)$, with $q=2^n+1\geq 5$.
Let $Q$ be a conjugacy closed loop, and $N(Q)$ its nucleus. Then $Z(N(Q))$ contains all associators of elements of $Q$. If in addition $Q$ is diassociative (i.e., an extra loop), then all these associators have order 2. If $Q$ is power-associative and $|Q|$ is finite and relatively prime to 6, then $Q$ is a group. If $Q$ is a finite non-associative extra loop, then $16 \mid |Q|$.
In this note we show that the commutant of a Bol loop of odd order is a subloop.
The purpose of this brief note is to sharpen a result of Kepka about the axiomization of the variety of trimedial quasigroups.
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 diassociative, thus generalizing Moufang's Theorem.
An A-loop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative A-loop is Moufang.