Semidirect product of weak inverse property power associative conjugacy closed loops

Abstract

This work provides a necessary and sufficient conditions for the semidirect product of loops to produce a weak inverse property power associative conjugacy closed loop. It is shown that if $ A $, $ K $ and $ Q $ are loops such that $Q=A\times_{\theta}K$, where $\theta:A\rightarrow \textrm{Sym}K.$ Then $ Q $ is weak inverse property power associative conjugacy closed loop if and only if $ A $ and $ K $ are weak inverse property power associative conjugacy closed loops, $\theta(x)$ is a nuclear automorphism of $K$ in the sense that $ k^{-1}k\theta(x)\in N(K)\;\mbox{for each}\; x\in A, \;k\in K $ and $\theta(xy)=\theta(x)\theta(y).$