Constructing (pre)norms in BL-algebras and its action on topological BL-algebras

Abstract

In this paper we define the notions of a norm and a prenorm on BL-algebras and we give some methods for constructing them. We introduce (pre)metrics on BL-algebras and find some connections between them and (pre)norms. Using prenorms, we find some conditions that a BL-algebra become a metric space. Finally, we show that if there exists a prenorm on a BL-algebra A, then for any filter Fin A, there are topologies U and Von A and on A/F respectively such that (A, U) and (A / F, V) are (semi)topological BL-algebras.