Loops with squares in two nuclei
Michael Kinyon, J. D. Phillips
TL;DR
The paper investigates the structure of loops whose squares lie in two nuclei (notably left-middle and left-right nuclear squares) and the broader class of loops with central squares. It leverages principal isostrophes to transfer results between loop varieties, proves normality results for intersections of nuclei such as $\mathrm{Nuc}_{\ell,m}(Q)$ and $\mathrm{Nuc}_{\ell,r}(Q)$, and characterizes central squares via the automorphic inverse property (AIP) or endomorphic squaring. A Decomposition Theorem is established for torsion loops in this class, showing $Q\cong E\times O$, with $E$ the 2-power part and $O$ the odd-order part, yielding a clear split of the torsion structure. Specialization to left C loops yields quotient Steiner loop behavior, simple left C loop classifications (groups or left Steiner loops), and a single-identity axiomatization for AIP-left-C loops. These results advance the structural understanding of Bol-Moufang type loops and connect left C loops to well-studied Steiner constructions.
Abstract
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 structure than one might initially guess. Loops with squares in (at least) two nuclei include many well known classes of loops, such as C loops and extra loops, and not so well known classes such left C loops. In any loop with, say, left and middle nuclear squares, the intersection of the left and middle nuclei is a normal subloop; hence such a loop is simple if and only if it is a group or a simple unipotent loop. Loops in which squaring is a centralizing endomorphism have even more structure; they are power-associative, and a torsion loop in that class is a direct product of a loop of $2$-elements and a loop of elements of odd order.