Poset product and BL-chains

CONRADO GOMEZ

Barcelona

Conferencia; Syntax meets semantics (SYSMICS); 2016

Basic Fuzzy Logic (BL for short) was introduced by Hájek in [Há98b] to formalize fuzzy logics in which the conjunction is interpreted by a continuous t-norm on the real segment [0; 1] and the implication by its corresponding adjoint. BL-algebras are the algebraic counterpart of BL. These algebras form the variety BL, which has many well known subvarieties, including the variety MV of MV-algebras (the algebraic semantics for Łukasiewicz?s logic), the variety ofProduct algebras and the variety of Gödel algebras. One of the most important properties of BL is that it is generated by its totally ordered members, called BL-chains. This reason explains why the first attempts to investigate BL focused on the structure of BL-chains. For instance, in [CEGT00] the authors characterized saturated BL-chains as ordinal sums of MV-chains, product chains and Gödel chains, resembling the famous Mostert-Schield decomposition for t-norms. Although this characterization cannot be extended to every BL-chain, it gives information on the structure of BL-chains in general,since each BL-chain can be isomorphically embedded in a saturated chain.Clearly, to have a representation theorem in terms of simpler or better known structures helps to understand BL-algebras.BL can also be seen as the subvariety of bounded basic hoops BH, i.e. bounded hoops that are isomorphic to subdirect products of totally ordered hoops. Thus, the general algebraic theory of hoops applies to BL-algebras as well. As Aglianò and Montagna noted in [AM03], the fundamental structures in the study of BL-algebras are Wajsberg hoops, whose structure are simpler than BL. They proved that each BL-chain can be uniquely decomposed into an ordinal sum of totally ordered Wajsberg hoops. It is worth to say that the definition for ordinal sumin [AM03] differs from the one given in [CEGT00]. However, none of the decompositions mentioned until here can be generalized to non totally ordered BL-algebras.The poset product is a construction introduced by Jipsen and Montagna in [JM09]. In a sense, the poset product generalizes the notions of ordinal sum and direct product. Given a poset P and a collection {Lp : p in P} of commutative residuated lattices sharing the same neutral element 1 and the same minimum element 0, the poset product is the lattice L defined as follows:1. The domain of L is the set of all x belonging to the direct product of Lp such that for all i in P, if xi is not 1, then xj = 0 provided that j > i.2. The monoid operation and the lattice operations are defined pointwise.3. The residual depends on the index set PBased on the results of [JM06], [JM09] and [JM10], one can find in [BM11, Theorem 3.5.4] a proof that every BL-algebra is a subdirect poset product of MV-chains and product chains indexed by a poset P which is a forest. Therefore each BL-algebra is a subalgebra of a poset product of MV-chains and product chains. Unfortunately, this embedding is not surjective in general, even when dealing with chains.Our work is framed in the study of BL-algebras that admit a representation as poset product. In this communication we will present a necessary and sufficient condition to establish when an ordinal sum of BL-chains coincides with the poset product of the same collection. From the study of BL-algebras that are not representable in this sense will arise the significant role that the index poset plays in the poset product construction. For instance, we will see that those BL-chains that are isomorphic to a poset product of MV-chains and product chains form a proper subset of the set of saturated BL-chains.References[AM03] P. Aglianò and F. Montagna. Varieties of BL-algebras I: general properties. Journal of Pure and Applied Algebra, 181:105?129, 2003.[BF00] W.J. Blok and I.M.A. Ferreirim. On the structure of hoops. Algebra Universalis, 43:233?257, 2000.[BM11] M. Busaniche and F. Montagna. Hájek?s logic BL and BL-algebras. In Handbook of Mathematical Fuzzy Logic, volume 1 of Studies in Logic, Mathematical Logic and Foundations, chapter V, pages 355?447. College Publications, London, 2011.[CDM99] R. Cignoli, I.M.L. D?Ottaviano, and D. Mundici. Algebraic Foundations of Many-Valued Reasoning, volume 7 of Trends in Logic. Kluwer Academic Publishers, Dordrecht, 1999.[CEGT00] R. Cignoli, F. Esteva, Ll. Godo, and A. Torrens. Basic fuzzy logic is the logic of continuous t-norm and their residua. Soft Computing, 4(2):106?112, 2000.[GJKO07] N. Galatos, P. Jipsen, T. Kowalski, and H. Ono. Residuated lattices: an algebraic glimpse at substructural logics, volume 151 of Studies in logic and the foundation of mathematics. Elsevier, Amsterdam, 2007.[Há98a] P. Hájek. Basic fuzzy logic and BL-algebras. Soft Computing, 2:124?128, 1998.[Há98b] P. Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer Academic Publishers, Dordrecht, 1998.[Jip09] P. Jipsen. Generalizations of boolean products for lattice-ordered algebras. Annals of Pure and Applied Logic, 161:228?234, 2009.[JM06] P. Jipsen and F. Montagna. On the structure of generealized BL-algebras. Algebra Universalis, 55:227?238, 2006.[JM09] P. Jipsen and F. Montagna. The Blok-Ferreirim theorem for normal GBL-algebras and its applications. Algebra Universalis, 60:381?404, 2009.[JM10] P. Jipsen and F. Montagna. Embedding theorems for classes of GBL-algebras. Journal of Pure and Applied Algebra, 214:1559?1575, 2010.