Semi–Boolean and Hyper–Archimedean BL–Algebras

Abstract

We analyze differences between BL–algebras and MV–algebras. The study has application in mathematical fuzzy logic as the Lindenbaum algebras of Lukasiewicz logic or Hájek’s BL–logics are MV–algebras or BL–algebras, respectively. We focus on possible generalizations of Boolean elements of a general BL–algebra L; we prove that an element x ∈ L is Boolean iff x ∨ x ^∗ = 1. L is called semi–Boolean if, for all x ∈ L, x ^∗ is Boolean. We prove that an MV–algebra L is semi–Boolean iff L is a Boolean algebra. A BL–algebra L is semi–Boolean iff L is a SBL–algebra. A BL–algebra L is called hyper–Archimedean if, for all x ∈ L, there is an n ≥ 1 such that x ⁿ is Boolean. We prove that hyper–Archimedean BL–algebras are MV–algebras. We discuss briefly the applications of our results in mathematical fuzzy logic.