SOME ALGEBRAIC PROPERTIES OF GENERALISED CENTRAL LOOPS

PROPERTIES OF GENERALISED CENTRAL LOOPS

Abstract

Generalised central loops ($GCL$) are loops satisfying the identity\\ $x(z\cdot z^{\sigma}y) = (xz\cdot z^{\sigma})y$. In this work, three generalised identities corresponding to three of the four left central identities are newly introduced and all of these three generalised identities together with identity $z(z^{\sigma}\cdot xy) = (z\cdot z^{\sigma}x)y$, which was introduced in \cite{ref12} are shown to be equivalent in any loop. It is shown that every $GCL$ $(G, \cdot, \sigma) = (G, \cdot)$ is a $\sigma-$central square loop. Furthermore, it is established that a loop $(G, \cdot, \sigma) = (G, \cdot)$ is a $GCL$ if and only if $L_{z^{\sigma}}L_z$ and $R_zR_{z^{\sigma}}$ are crypto-automorphisms of $(G, \cdot, \sigma) = (G, \cdot)$ with companions $c_1 = (zz^{\sigma})^{-1}$ and $c_2 = e$, and companions $d_1 = e$ and $d_2 = (zz^{\sigma})^{-1}$ respectively. The necessary and sufficient conditions for a $GCL$ to be isomorphic to its principal isotopes are also formulated. Every pseudo-automorphism of a $GCL$ $(G, \cdot, \sigma) = (G, \cdot)$ with companion $zz^{\sigma}$ is shown to be a semi-automorphism. Lastly, a $GCL$ was constructed using a group together with an arbitrary subgroup of it.