A family of poset product-representable BL-algebras

CONRADO GOMEZ

Cagliari

Conferencia; AsubL - "Algebra and Substructural Logics" (take 6); 2018

The variety BL of BL-algebras is the algebraic counterpart of BL (Hájek's Basic Fuzzy Logic [4]). A BL-algebra is a divisible and commutative residuated lattice which is also prelinear. Among others, the varieties of MV-algebras and product algebras are well-known subvarieties of BL. Due to the prelinerity property, the fundamental structures in the study of BL are its totally orderedmembers (BL-chains). Focused on BL-chains, it was proved that they can becompletely described as an ordinal sum (of simpler structures).Theorem 1 (Subdirect representation theorem. See [4]). Each BL-algebra is a subdirect product of BL-chains.Theorem 2 (Decomposition theorem. See [1]). Each non-trivial BL-chain admits (up to isomorphism) a unique decomposition into an ordinal sum of non-trivial totally ordered Wajsberg hoops.Since every BL-algebra can be embedded into the direct product of BL-chains and every BL-chain can be decomposed as an ordinal sum, Jipsen and Montagna proposed in [6] a construction called poset product as a sort of generalization of direct product and ordinal sum. Briefly, the poset product is a subset of a direct product which is defined by using a partial order over the index set.In [3], based on the results of [5,6,7], it is shown that every BL-algebra can be thought as a subalgebra of the poset product of a collection of BL-chains.Theorem 4 (See [3]). Every BL-algebra can be embedded into a poset product of a family of MV-chains and product chains indexed by a forest.Hence it is natural to wonder if a BL-algebra is isomorphic to a poset product. Although finite BL-algebras are indeed isomorphic to a poset product of MV-chains (details in [6]), in general the answer is negative (even for BL-chains, as shown in [2]).Our work is framed in the study of BL-algebras that are isomorphic to a poset product of BL-chains. The aim of this talk is to examine some features of this construction and consider the restriction referred above. Then we will introduce the notions of indecomposable and representable BL-algebra in the sense of poset product. Our main result provides sufficient conditions for BL-algebras so that they admit a representation as a poset product. The proof, which is constructive, will be outlined.Theorem 5. Let A be a BL-algebra. If its subset J(A) of idempotent and join irreducible elements is a well partial order (with the inherited order) such that each i ∈ J(A) induces a prime filter in A, and every a ∈ A has a maximum idempotent element below it, then A is isomorphic to the poset product of a collection of indecomposable BL-chains.References[1] P. Aglianò and F. Montagna. Varieties of BL-algebras I: general properties. Journal of Pure and Applied Algebra, 181:105-129, 2003.[2] M. Busaniche and C. Gomez. Poset product and BL-chains. Studia Logica, 2017. https://doi.org/10.1007/s11225-017-9764-6.[3] 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.[4] P. Hájek. Metamathematics of Fuzzy Logic. Trends in Logic, Volume 4. Kluwer Academic Publishers. 1998.[5] P. Jipsen, Generalizations of Boolean products for lattice-ordered algebras. Annals of Pure and Applied Logic, 161:228-234, 2009.[6] P. Jipsen and F. Montagna. The Blok-Ferreirim theorem for normal GBL-algebras and its applications. Algebra Universalis, 60:381-404, 2009.[7] P. Jipsen and F. Montagna. Embedding theorems for classes of GBL-algebras. Journal of Pure and Applied Algebra, 214:1559-1575, 2010.