Louie Operator

Back in April 2015, during a retreat, I attempted to "make" my own operator. I'm pretty sure that a similar operator has been done. Why did I use the "heart"? I don't really know... I remember doing this during the communion hour, so maybe the "heart" symbol can mean the love of Christ. It can also mean romantic love. Anyways, from its definition, I started exploring its properties. From its properties, I started exploring theorems. Currently, I don't see errors yet but maybe others can see more. ：D