boolean element

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.

In this paper, we study residuated lattices in order to give new characterizations for dense, regular and Boolean elements in residuated lattices and investigate special residuated lattices in order to obtain new characterizations for the directly indecomposable subvariety of Stonean residuated lattices. Free algebra in varieties of Stonean residuated lattices is constructed. We introduce in residuated lattice a new type of filter called special filter and investigate its properties. Finally, regular filter property in residuated lattices is introduced and is studied in details.

In this paper, some new properties of EQ -algebras are investigated. We introduce and study the notion of Boolean center of lattice ordered EQ-algebras with bottom element. We show that in a good ℓEQ-algebra E with bottom element the complement of an element is unique. Furthermore, Boolean elements of a good bounded lattice EQ-algebra are characterized. Finally, we obtain conditions under which Boolean center of an EQ-algebra E is the subalgebra of E