6 research outputs found
The variety generated by all the ordinal sums of perfect MV-chains
We present the logic BL_Chang, an axiomatic extension of BL (see P. H\'ajek -
Metamathematics of fuzzy logic - 1998, Kluwer) whose corresponding algebras
form the smallest variety containing all the ordinal sums of perfect MV-chains.
We will analyze this logic and the corresponding algebraic semantics in the
propositional and in the first-order case. As we will see, moreover, the
variety of BL_Chang-algebras will be strictly connected to the one generated by
Chang's MV-algebra (that is, the variety generated by all the perfect
MV-algebras): we will also give some new results concerning these last
structures and their logic.Comment: This is a revised version of the previous paper: the modifications
concern essentially the presentation. The scientific content is substantially
unchanged. The major variations are: Definition 2.7 has been improved.
Section 3.1 has been made more compact. A new reference, [Bus04], has been
added. There is some minor modification in Section 3.
Priestley duality for MV-algebras and beyond
We provide a new perspective on extended Priestley duality for a large class
of distributive lattices equipped with binary double quasioperators. Under this
approach, non-lattice binary operations are each presented as a pair of partial
binary operations on dual spaces. In this enriched environment, equational
conditions on the algebraic side of the duality may more often be rendered as
first-order conditions on dual spaces. In particular, we specialize our general
results to the variety of MV-algebras, obtaining a duality for these in which
the equations axiomatizing MV-algebras are dualized as first-order conditions
Soju Filters in Hoop Algebras
The notions of (implicative) soju filters in a hoop algebra are introduced, and related properties are investigated. Relations between a soju sub-hoop, a soju filter and an implicative soju filter are discussed. Conditions for a soju filter to be implicative are displayed, and characterizations of an implicative soju filters are considered. The extension property of an implicative soju filter is established
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group
Algebraic study of axiomatic extensions of triangular norm based fuzzy logics
According to the Zadeh’s famous distinction, Fuzzy Logic in narrow sense, as
opposed to Fuzzy Logic in broad sense, is the study of logical systems aiming
at a formalization of approximate reasoning. In the systems commonly used the
strong conjunction connective is interpreted by a triangular norm (t-norm, for
short) while the implication connective is interpreted by its residuum. Therefore,
the usual logical systems for Fuzzy Logic are based on t-norms with a residuum.
The necessary and sufficient condition for a t-norm to have a residuum is the
left-continuity. In order to define the based t-norm based fuzzy logic, Esteva and
Godo introduced the system MTL, which was indeed proved to be complete with
respect to the semantics given by all left-continuous t-norms and their residua.
In the first part of this dissertation we have carried out an attempt to describe
the axiomatic extensions of MTL, paying special attention to those which are also
t-norm based. We have done it from an algebraic point of view, by exploiting the
fact that these logics are algebraizable by varieties of MTL-algebras. Therefore,
our study has resulted in an algebraic study of such varieties, where the final
aim would be to obtain a description of the structure of their lattice and their
relevant properties. Although this description has not been achieved yet, we
have done several significant advances in this direction that can be classified in
two groups: (a) those that spread some light over the amazing complexity of the
lattice, and (b) those that describe some well-behaved parts of the lattice. More
precisely:
• By considering the connected rotation-annihilation method proposed to
build involutive left-continuous continuous t-norm, we have proposed a
possible way to decompose MTL-chains and we have studied some particular cases of this decomposition. This has resulted in an extension of the
theory of perfect, local and bipartite algebras formerly used in varieties of
MV and BL-algebras, to the variety of all MTL-algebras.
• Perfect IMTL-algebras have been proved to be exactly (module isomorphism) the disconnected rotations of prelinear semihoops (a particular case
of the decomposition as connected rotation-annihilation).
• The lattice of varieties generated by perfect IMTL-algebras has been
proved to be isomorphic to the lattice of varieties of prelinear semihoops.
xiii
xiv
• A decomposition theorem of every MTL-chain as an ordinal sum of indecomposable prelinear semihoops has been obtained. Since all IMTL-chains
are indecomposable and, as the previous item states, we have the complexity of all the lattice of varieties inside the involutive part, the description
of all indecomposable prelinear semihoops seems to be a hopeless task.
• A particular class of indecomposable MTL-chains has been studied, namely
weakly cancellative chains. We have studied the logics associated to these
chains.
• We have studied the varieties of MTL-chains where a weak form of contraction, the so-called n-contraction law, holds. This condition yields a
global form of Deduction Detachment Theorem and allows to prove several properties of their related logics.
• We have focused on a particular subvariety of 3-contractive MTL-algebras,
namely Weak Nilpotent Minimum algebras, obtaining a number of results
on axiomatization of their subvarieties, local finiteness, generic chains and
standard completeness.
In the second part of the dissertation we consider another significant question
of Fuzzy Logic: which should be the use of the intermediate truth-values? In
MTL and its extensions, these truth-values for partial truth do not seem to be
used in a deep way, since in the algebraization of the logics the only distinguished
value is the top element. Following Pavelka’s idea, we consider expansions of tnorm based logics with constants for the intermediate truth-values, allowing
them to play an explicit role in the language. The originality of our proposal
lies in carrying out an algebraic approach to these expansions and studying their
standard completeness propertie