Mate I Loops [Articol]

Abstract

A loop (Q; ) is called a Mate I loop if it obeys the identity (x xy)z = (yz x)x. Interestingly, a Mate I loop is a left central loop (LC-loop) but not necessarily a central loop or an extra loop (and vice versa). In this paper, it is shown that a loop (Q; ) is a Mate I loop if and only if it is a LC-loop and a 2can IV loop. Furthermore, we show that a centrum square loop is a Mate I loop if and only if it is a central loop. A Mate I loop is also shown to be a 2can IV loop, a Triad IV loop and a Triad VII loop. Its relationship with some other loops of perfect type is established. Additionally, we proved that a left Bol loop that is also a left Cheban loopis a Mate I loop, and in a Mate I loop, the left Bol law, the LCC law and the left Cheban law are equivalent. Importantly, a fascinating discovery in this study is the fact that a Mate I loop was found to possess two particular properties which two new classes of power associative left (right) conjugacy closed loops possess (each characterizes weak inverse property, power associative conjugacy closed loop).