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ALGEBRAIC AND TOPOLOGICAL METHODS IN NON-CLASSICAL LOGICS III (TANCL'07)
August 5-9, 2007
St Anne's College, University of Oxford
Oxford, England

Organizers
Mai Gehrke and Hilary Priestley

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Semisimplicity, EDPC and discriminator varieties of bounded commutative residuated lattices with S4-like modal operator
by
Hiroki Takamura
National Institute of Advanced Industrial Science and Technology(AIST)/Research Center for Verification and Semantics(CVS)

Abstract

Substructural logics are logics lacking some or all of the structural rules when they are formalized in sequent systems. The FL of full Lambek calculus is obtained from the Gentzen system LJ for intuitionistic propositional logic by removing three structural rules: the rules of exchange, weakening and contraction. The algebraic structures for substructural logics are called residuated lattices that have been studied by algebraists since the 1930s, but the study has been revived recently as a study of mathematical structures for substructural logics. More information about residuated lattices, see the references [Kowalski-Ono2001, Jipsen-Tsinakis2002], and the relations residuated lattices and substructural logics, see the references [Galatos-Jipsen-Kowalski-Ono2007, Kowalski-Ono2001]. Algebraic study of substractural logics with modality (modal substructural logics) is first introduced by H. Ono. There are papers about algebraic study of modal substructural logics, however this field is not so much developed yet.

In this paper we mention that all semisimple varieties of bounded commutative residuated lattices with S4-like modal operator are discriminator varieties. We also give a characterization of discriminator and EDPC (equational definable principal congruence) varieties of commutative residuated lattices with modal operator follows.

In [Kowalski2004], the author proves that all semisimple varieties of FLew-algebras which is the algebraic structure for one of substructural logics, FLew, are discriminator varieties. In this paper, his proof works well also for the case of the BCRL[¯], with some modification.

A bounded commutative residuated lattice with S4-like modal operator (BCRL[¯]) is a structure A=〈A, ∧, ∨, ·, →, T, ⊥, 1, [¯]〉, such that:

(1) 〈A, ∧, ∨, ·, →, T, ⊥, 1〉 is a bounded commutative residuated lattice with the greatest element T, the least element ⊥ and monoid unit 1,
(2) [¯] is a unary operation on A satisfying: (a) 1 ≤ [¯] 1, (b) [¯] x ·[¯] y ≤ [¯] (x ·y), (c) [¯] x ≤ x, (d) [¯] x ≤ [¯][¯] x, (e) if x ≤ y then [¯] x ≤ [¯] y for any x, y ∈ A.

Theorem For a variety V of BCRL[¯], the following conditions are equivalent:
(i) V satisfies [¯] (x∧1) ∨¬([¯] (x∧1))n, for some natural number n;
(ii) V is semisimple;
(iii) V is a discriminator variety.

[Kowalski-Ono2001] T. Kowalski and H. Ono, Residuated lattices: an algebraic glimpse at logics without contraction, JAIST preliminary report 2001.

[Jipsen-Tsinakis2002 P. Jipsen and C. Tsinakis, A survey of residuated lattices, in Ordered Algebraic Structures (ed J. Martines) Kluwer Academic Publishers, Dordrecht 2002.

[Galatos-Jipsen-Kowalski-Ono2007] N. Galatos, P. Jipsen, T. Kowalski and H. Ono, Residuated lattices: an algebraic glimpse at logics without contraction, (Studies in Logic and the Foundations of Mathematics, vol. 151), Elsevier, 2007

[Kowalski2004] T. Kowalski, Semisimplicity, EDPC and discriminator varieties of residuated lattices, Studia Logica 77, 2004, no. 2, 255-265.

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Date received: May 14, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caug-53.