Loops with squares in two nuclei

TL;DR

This paper investigates loops with squares in multiple nuclei, revealing their structural properties, classifications, and conditions under which they are simple or decomposable, with implications for well-known loop classes.

Contribution

It characterizes loops with squares in two nuclei, showing their structural properties and conditions for simplicity and decomposition, expanding understanding of loop classifications.

Findings

Loops with squares in two nuclei include C and extra loops.

The intersection of left and middle nuclei is a normal subloop.

Loops with centralizing square endomorphisms are power-associative and decomposable.

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.