DM-GAA

In this article, some algebraic characterisations of generalised middle Bol loop (GMBL) using its parastrophes and holomorph were studied. In particular, it was shown that if the generalised map $\alpha$ is bijective such $\alpha: e\rightarrow e$, then the $(12)-$ parastrophe of GMBL is a GMBL. The conditions for $(13)-$ and $(123)-$parastrophes of a GMBL to be GMBL of exponent two were unveiled. We further established that a commutative $(13)-$ and $(123)-$parastrophes of GMBL has an inverse properties. $(23)-$ parastrophe of $Q$ was shown to be super $\alpha-$elastic property if it has a middle symmetric while $(132)-$parastrophe of $Q$ satisfies left $ \alpha-$symmetric. It is further shown that a commutative $(13)-$ and $(123)-$ parastrophes of $Q$ are generalised Moufang loops of exponent two. Also, commutative $(132)-$ and $(23)-$ parastophes of $Q$ are shown to be Steiner loops. A necessary and sufficient condition for holomorph of generalised middle Bol loop to be GMBL was presented. The holomorph of a commutative loop was shown to be a commutative generalised middle Bol loop if and only if the loop is a GMBL.

Keywords:

parastrophe , GMBL, holomorph

References:

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