Global Left Loop Structures on Spheres

On the unit sphere $\mathbb{S}$ in a real Hilbert space $\mathbf{H}$, we derive a binary operation $\odot$ such that $(\mathbb{S},\odot)$ is a power-associative Kikkawa left loop with two-sided identity $\mathbf{e}_0$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot$ is compatible with the symmetric space structure of $\mathbb{S}$. $(\mathbb{S},\odot)$ is not a loop, and the right translations which fail to be injective are easily characterized. $(\mathbb{S},\odot)$ satisfies the left power alternative and left Bol identities ``almost everywhere'' but not everywhere. Left translations are everywhere analytic; right translations are analytic except at $-\mathbf{e}_0$ where they have a nonremovable discontinuity. The orthogonal group $O(\mathbf{H})$ is a semidirect product of $(\mathbb{S},\odot)$ with its automorphism group (cf. http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of $(\mathbb{S},\odot)$ gives some insight into spherical geometry.Comment: 18 pages, no figures, 10pt, LaTeX2e, uses amsart.cls & tcilatex.tex. To appear in Comment. Math. Univ. Carolin. (special issue: Proceedings of LOOPS99) Revised version: various fixes and improvements suggested by refere

Infinite Simple Bol Loops

If the left multiplication group of a loop is simple, then the loop is simple. We use this observation to give examples of infinite simple Bol loops.Comment: 4 pages, AMS-LaTeX, to appear in Comment. Math. Univ. Carolinae for a special issue: the Proceedings of Loops03. Version 3: more minor changes suggested by the refere

Axioms for trimedial quasigroups

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.Comment: 6 pages, AMS-LaTeX. To appear in Comment. Math. Univ. Carolinae. for a special issue: the Proceedings of Loops03. Version 3: the proof of the main result is collected together more formally; other stylistic change