Hajek basic fuzzy logic and Lukasiewicz infinite-valued logic

Using the theory of BL-algebras, it is shown that a propositional formula phi is derivable in Lukasiewicz infinite valued Logic if and only if its double negation similar tosimilar tophi is derivable in Hajek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (phi (phi double right arrow similar to phi)) double right arrow psi then phi is derivable in in classical logic if and only if similar tosimilar tophi is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras.