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On the reflection invariance of residuated chains

Sándor Jenei

Annals of Pure and Applied Logic 161 (2):220-227 (2010)

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Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops.Carles Noguera, Francesc Esteva & Joan Gispert - 2005 - Archive for Mathematical Logic 44 (7):869-886.details IMTL logic was introduced in [12] as a generalization of the infinitely-valued logic of Lukasiewicz, and in [11] it was proved to be the logic of left-continuous t-norms with an involutive negation and their residua. The structure of such t-norms is still not known. Nevertheless, Jenei introduced in [20] a new way to obtain rotation-invariant semigroups and, in particular, IMTL-algebras and left-continuous t-norm with an involutive negation, by means of the disconnected rotation method. In order to give an algebraic interpretation (...) Download Export citation Bookmark 10 citations
On the continuity points of left-continuous t-norms.S. Jenei & F. Montagna - 2003 - Archive for Mathematical Logic 42 (8):797-810.details Left-continuous t-norms are much more complicated than the continuous ones, and obtaining a complete classification of them seems to be a very hard task. In this paper we investigate some aspects of left-continuous t-norms, with emphasis on their continuity points. In particular, we are interested in left-continuous t-norms which are isomorphic to t-norms which are continuous in the rationals. We characterize such a class, and we prove that it contains the class of all weakly cancellative left-continuous t-norms. Download Export citation Bookmark 2 citations
Kripke semantics, undecidability and standard completeness for Esteva and Godo's logic MTL∀.Franco Montagna & Hiroakira Ono - 2002 - Studia Logica 71 (2):227-245.details The present paper deals with the predicate version MTL of the logic MTL by Esteva and Godo. We introduce a Kripke semantics for it, along the lines of Ono''s Kripke semantics for the predicate version of FLew (cf. [O85]), and we prove a completeness theorem. Then we prove that every predicate logic between MTL and classical predicate logic is undecidable. Finally, we prove that MTL is complete with respect to the standard semantics, i.e., with respect to Kripke frames on the (...) Download Export citation Bookmark 12 citations
A proof of standard completeness for Esteva and Godo's logic MTL.Sándor Jenei & Franco Montagna - 2002 - Studia Logica 70 (2):183-192.details In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in. Download Export citation Bookmark 30 citations
On the structure of rotation-invariant semigroups.Sándor Jenei - 2003 - Archive for Mathematical Logic 42 (5):489-514.details We generalize the notions of Girard algebras and MV-algebras by introducing rotation-invariant semigroups. Based on a geometrical characterization, we present five construction methods which result in rotation-invariant semigroups and in particular, Girard algebras and MV-algebras. We characterize divisibility of MV-algebras, and point out that integrality of Girard algebras follows from their other axioms. Download Export citation Bookmark 18 citations
On the standard and rational completeness of some axiomatic extensions of the monoidal t-Norm logic.Francesc Esteva, Joan Gispert, Lluís Godo & Franco Montagna - 2002 - Studia Logica 71 (2):199 - 226.details The monoidal t-norm based logic MTL is obtained from Hájek''s Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (...) Download Export citation Bookmark 19 citations

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