Right-Continuous Mappings and Fuzzy Ideals

Abstract

We introduce the concept of right-continuous mappings from [0,1) to the power set of finite elements in a complete semilattice and define a closure operator on the set $P_r$ of right-continuous mappings. We show that the lattice of fuzzy ideals on the semilattice of finite elements is isomorphic to the lattice of closed elements concerning the closure operator. Furthermore, we give equivalent conditions for regular closure operators on the set of right-continuous mappings to be quasi-algebraic. Finally, we prove the lattice of fuzzy ideals on the set of finite elements in an algebraic semilattice is subdirectly embedded into the product of copies of the semilattice, and show the subdirect embedding has a universal mapping property.