BL-algebras and effect algebras

Abstract

Although the notions of a BL-algebra and of an effect algebra arose in rather different contextes, both types of algebras have certain structural properties in common. To clarify their mutual relation, we introduce weak effect algebras, which generalize effect algebras in that the order is no longer necessarily determined by the partial addition. A subclass of the weak effect algebras is shown to be identifiable with the BL-algebras. Moreover, weak D-posets are defined, being based on a partial difference rather than a partial addition. They are equivalent to weak effect algebras. Finally, it is seen to which subclasses of the weak effect algebras certain subclasses of the BL-algebras, namely the MV-, product, and Gödel algebras, correspond.