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This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

In this paper we prove that Basic Logic (BL) is complete w.r.t. the continuous t-norms on [0,1], solving the open problem posed by Ha Âjek in [4]. In fact, Ha Âjek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axioms of BL. The main result of the paper is to show that B1 and B2 axioms are indeed redundant. We also obtain an improvement of the decomposition theorem for saturated BL-chains as ordinal sums whose components are either MV, product or Go Èdel chains, in an analogous way as for continuous t-norms. Finally we provide equational characterizations of the variety of BL-algebras generated by the three basic BL subvarieties, as well as of the varieties generated by each pair of them, together with completeness results of the calculi corresponding to all these subvarieties.Key words many-valued logic (basic, èukasiewicz, product and Go Èdel logics), equational classes, fuzzy logic, t-norms and standard completeness IntroductionBasic Logic (BL for short) and the corresponding algebras (BL-algebras) were introduced by Ha Âjek (see [3] and the references given there) as an attempt to axiomatize the many-valued semantics induced by continuous t-norms on the unit real interval [0, 1].As a ®rst step, Ha Âjek showed that a propositional formula is provable in BL if and only if it is a tautology in any linearly ordered BL-algebra (BL-chain for short). However, completeness of BL w.r.t. the BL algebras in [0, 1] induced by continuous t-norms was left as an open problem in [3].In the recent paper [4], Ha Âjek proved that such completeness theorem can be obtained provided two new axioms, B1 and B2, were added to the original axiomatic system of BL. These new axioms are rather unnatural and without a clear logical meaning. Hence it is natural to ask, as Ha Âjek did, whether they are necessary or can be derived from the original axioms. The aim of this paper is to solve this problem, showing that the new axioms are indeed redundant, i.e., that the original axiomatic of BL is complete w.r.t. the continuous t-norms on [0, 1].After a summary in Sect. 2 of main results of [4], Sect. 3 deals with the structure of BL-chains. Axiom B1 was introduced to prevent the existence of some``pathological triples'' in saturated and irreducible BL-chains. We show (Theorem 3.1) that these pathological triples cannot exist in any BL-chain, without using axiom B1. The role of Axiom B2 was to guarantee that a saturated and irreducible BL-chain with zero divisors is an MV-algebra. We present two proofs of this fact (Lemma 3.3 and Remark 2), neither of them depending on B2. In fact, in Theorem 3.4 we show that any saturated and irreducible BL-chain is either a MV chain or a product chain. In this way we obtain an improvement of the decomposition theorem for saturated Theorem 4]: any saturated BL-chain is an ordinal sum whose components are either MV, product or Go Èdel chains. The analogy between this representation and the well-known decomposition theorem for continuous t-no...