On the universality of some Smarandache loops of Bol-Moufang type

Abstract

A Smarandache quasigroup(loop) is shown to be universal if all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Also, weak Smarandache loops of Bol-Moufang type such as Smarandache: left(right) Bol, Moufang and extra loops are shown to be universal if all their f,g-principal isotopes are Smarandache f,g-principal isotopes. Conversely, it is shown that if these weak Smarandache loops of Bol-Moufang type are universal, then some autotopisms are true in the weak Smarandache sub-loops of the weak Smarandache loops of Bol-Moufang type relative to some Smarandache elements. Futhermore, a Smarandache left(right) inverse property loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes is shown to be universal if and only if it is a Smarandache left(right) Bol loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Also, it is established that a Smarandache inverse property loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes is universal if and only if it is a Smarandache Moufang loop in which all its f,g-principal isotopes are Smarandache f,g-principal isotopes. Hence, some of the autotopisms earlier mentioned are found to be true in the Smarandache sub-loops of universal Smarandache: left(right) inverse property loops and inverse property loops.