On three implication-less fragments of t-norm based fuzzy logics

The study of the Gentzen system Gew determined by the well known sequent calculus FLew [Ono98, Ono03c] is interesting for the study of the substructural aspects of t-norm based logics [Háj98, EG01]. In [BGV05] we studied the 〈∨, ∗,¬, 0, 1〉 and the 〈∨,∧, ∗,¬, 0, 1〉-fragments of this Gentzen system and the same fragments of the logic of residuated lattices IPC∗\c. In this paper we continue the study of the implication-less fragments of IPC∗\c and of Gew. We obtain the algebraization of the 〈∨, ∗, 0, 1〉 ,and the 〈∨,∧, ∗, 0, 1〉-fragment of Gew. We obtain also that 〈∨, ∗, 0, 1〉 and the 〈∨,∧, ∗, 0, 1〉-fragments of the logic of residuated lattices, IPC∗\c, are exactly the 〈∨,∧, 0, 1〉-fragment of classical logic. As a corollary of this fact we have that the 〈∨, ∗, 0, 1〉 and the 〈∨,∧, ∗, 0, 1〉-fragments of every t-norm based logic are exactly the same fragments of classical logic.