2025/10/22 by Michael Kinyon, Kinyon, Michael, J. D. Phillips +1
Mathematics · #20N05 #Algebraic and Geometric Analysis #FOS: Mathematics #Group Theory (math.GR) #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2510.19961
openalex publication_date 2025/10/22 · openalex created_date 2025/10/25 · openalex updated_date 2026/07/28
Although little can be gleaned about a loop with the property that its squares are, say, left nuclear (xx⋅ yz = (xx⋅ y)z), if its squares are also, say, middle nuclear ((x⋅ yy)z = x(yy⋅ 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.