A FURTHER INVESTIGATION ON THE CORE OF MIDDLE BOL LOOPS

ON THE CORE OF MIDDLE BOL LOOPS

Abstract

In this paper, further investigation of the core of middle Bol loop relative to the core of right Bol loop, is presented. The efforts revealed that, (i) if $T$ is any permutation of $Q$, then $xT\otimes yT=(x\oplus y)T$ if and only if $[(x\circ y\backslash\backslash x)T\circ f]I= (xT\circ f)I\circ(yT\circ f)I\backslash\backslash(xT\circ f)I, \forall\ x,y\in Q \textrm{~and some } f\in Q$, where $(Q,\oplus)$ and $(Q,\otimes)$ are respectively, the cores of the middle Bol loop $(Q,\circ)$ and its isotope $(Q,\ast)$. (ii) in particular it was also shown that a middle Bol loop satisfies the identity: $(x\circ f^{-1}\backslash\backslash x)\circ f=(x\circ f)\circ(x\circ f), \forall\ f,x\in Q$ if and only if for each isotope $(Q,\ast)$ of the middle Bol loop $(Q,\circ)$ given by $x\ast y=(xR^o_fT\circ yI)R^{o-1}_fI$, then $x\otimes y=x\oplus y, \forall\ x,y\in Q$, where $(Q,\otimes)$ and $(Q,\oplus)$ the cores of $(Q,\ast)$ and $(Q,\circ)$ respectively . (iii) it was shown also that, the core exhibits some left self symmetry, left self distributive and that, the middle Bol loop is right distributive over its core. It was also remarked that, the core of middle Bol loop exhibits a form of semi-automorphism. New results were obtained and old ones extended.