Normality, nuclear squares and Osborn identities

TL;DR

This paper investigates the properties of Osborn loops, showing they have a normal nucleus coinciding with other nuclei, and explores their connections to Moufang and CC loops, with implications for nuclear identities.

Contribution

It introduces identities of Osborn loops via nuclear identification and characterizes loops that are both Buchsteiner and Osborn.

Findings

Osborn loops have a normal nucleus coinciding with all nuclei.

In LC loops, the left and middle nucleus coincide and are normal.

Loops that are both Buchsteiner and Osborn have all squares in the nucleus.

Abstract

Let $Q$ be a loop. If $S\leq Q$ is such that $\varphi(S) \subseteq S$ for each standard generator of $\mathrm{Inn}(Q)$, then $S$ does not have to be a normal subloop. In an LC loop the left and middle nucleus coincide and form a normal subloop. The identities of Osborn loops are obtained by applying the idea of nuclear identification, and various connections of Osborn loops to Moufang and CC loops are discussed. Every Osborn loop possesses a normal nucleus, and this nucleus coincides with the left, the right and the middle nucleus. Loops that are both Buchsteiner and Osborn are characterized as loops in which each square is in the nucleus.