Ultra deductive systems and (nilpotent) Boolean elements in hoops

Document Type : Original Article


1 Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

2 Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea

3 Hatef Higher Education Institute, Zahedan, Iran


In this paper, first we define the concept of nilpotent element on a hoop H, study some properties of them and investigate the relation with ultra deductive systems. Then by using this notion, we introduce cyclic hoops and prove that every cyclic hoop has a unique generator and is a local MV-algebra. In the follows, we introduce the notion of Boolean elements on hoops and investigate some of their properties and relation among Boolean elements with ultra deductive systems and nilpotent elements. Finally, we introduce a functor between the category of hoops and category of Boolean elements of them.