Loops with squares in two nuclei

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.