On the relationship between fuzzy subgroups and indistinguishability operators

Fuzzy subgroups are revisited considering their close relationship with indistinguishability operators (fuzzy equivalences) invariant under translations. Different ways to obtain new fuzzy subgroups from a given one are provided and different ways to characterize normal fuzzy subgroups are obtained. The idea of double coset of two (crisp) subgroups allow us to relate them via their equivalence classes. This is generalized to the fuzzy framework. The conditions in which a fuzzy relation R on a group G can be considered a fuzzy subgroup of G × G are obtained.Peer ReviewedPostprint (author's final draft

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

Del Operador Apertura en la Matemática Morfológica Difusa

Context: Interior operators have interesting properties that can be used in detecting relevant features in digital images. In this respect, it is pertinent to study the behaviour of the opening and erosion operators from the perspective of fuzzy morphological mathematics.Method: Bearing in mind the reticular properties inherent to the mathematical anaysis of an image by fuzzy morphological mathematics we seek to find restrictions on the structural element intended to filter the image so as to obtain characterizations of the opening and erosion operators.Results: We prove that if the structural relationship is reflexive or ⇤ -antitransitive the opening operator is interior. On the other hand, we found that if the relationship meets both erosion is a interior operator.Conclusions: The fuzzy morphological operators can give us more information than the classical methods when we filter an image, especially if we implement the opening operator with a right structural element.Contexto: las propiedades que posee un operador de interior se pueden trasladar a la determinación de características específicas de imágenes, las cuales, desde la matemática morfológica difusa pueden llegar a ser analizadas por medio del operador erosión y apertura, al conjugar estas ideas es pertinente indagar en torno a la naturaleza de estos operadores.Método: gracias a las propiedades reticulares que posee el análisis matemático de una imagen por medio de matemática morfológica difusa, se busca dotar de restricciones al elemento estructural con el cual se desea filtrar la imagen para así obtener caracterizaciones del operador apertura y erosión.Resultados: se demuestre que si la relación estructural es reflexiva o -antitransitiva el operador apertura es interior; en caso que la relación cumpla ambas, la erosión es un operador interior.Conclusiones: los operadores morfológicos difusos permiten obtener información relevante sin alterar la estructural global de la imagen; arrojando mayor calidad que los métodos clásicos, en especial si se emplea el operador apertura difuso con una relación estructural adecuada

Logical and algebraic structures from Quantum Computation

The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures

Proceedings of the International Workshop "Innovation Information Technologies: Theory and Practice": Dresden, Germany, September 06-10.2010

This International Workshop is a high quality seminar providing a forum for the exchange of scientific achievements between research communities of different universities and research institutes in the area of innovation information technologies. It is a continuation of the Russian-German Workshops that have been organized by the universities in Dresden, Karlsruhe and Ufa before. The workshop was arranged in 9 sessions covering the major topics: Modern Trends in Information Technology, Knowledge Based Systems and Semantic Modelling, Software Technology and High Performance Computing, Geo-Information Systems and Virtual Reality, System and Process Engineering, Process Control and Management and Corporate Information Systems

Permutable fuzzy consequence and interior operators and their connection with fuzzy relations

Fuzzy operators are an essential tool in many fields and the operation of composition is often needed. In general, composition is not a commutative operation. However, it is very useful to have operators for which the order of composition does not affect the result. In this paper, we analyze when permutability appears. That is, when the order of application of the operators does not change the outcome. We characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators. We prove that for these cases, permutability is completely connected to the preservation of the operator type.; We also study the particular case of fuzzy operators induced by fuzzy relations through Zadeh's compositional rule and the inf--> composition. For this cases, we connect permutability of the fuzzy relations (using the sup-* composition) with permutability of the induced operators. Special attention is paid to the cases of operators induced by fuzzy preorders and similarities. Finally, we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces.Peer Reviewe

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL , in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established

Beyond shared memory loop parallelism in the polyhedral model

2013 Spring.Includes bibliographical references.With the introduction of multi-core processors, motivated by power and energy concerns, parallel processing has become main-stream. Parallel programming is much more difficult due to its non-deterministic nature, and because of parallel programming bugs that arise from non-determinacy. One solution is automatic parallelization, where it is entirely up to the compiler to efficiently parallelize sequential programs. However, automatic parallelization is very difficult, and only a handful of successful techniques are available, even after decades of research. Automatic parallelization for distributed memory architectures is even more problematic in that it requires explicit handling of data partitioning and communication. Since data must be partitioned among multiple nodes that do not share memory, the original memory allocation of sequential programs cannot be directly used. One of the main contributions of this dissertation is the development of techniques for generating distributed memory parallel code with parametric tiling. Our approach builds on important contributions to the polyhedral model, a mathematical framework for reasoning about program transformations. We show that many affine control programs can be uniformized only with simple techniques. Being able to assume uniform dependences significantly simplifies distributed memory code generation, and also enables parametric tiling. Our approach implemented in the AlphaZ system, a system for prototyping analyses, transformations, and code generators in the polyhedral model. The key features of AlphaZ are memory re-allocation, and explicit representation of reductions. We evaluate our approach on a collection of polyhedral kernels from the PolyBench suite, and show that our approach scales as well as PLuTo, a state-of-the-art shared memory automatic parallelizer using the polyhedral model. Automatic parallelization is only one approach to dealing with the non-deterministic nature of parallel programming that leaves the difficulty entirely to the compiler. Another approach is to develop novel parallel programming languages. These languages, such as X10, aim to provide highly productive parallel programming environment by including parallelism into the language design. However, even in these languages, parallel bugs remain to be an important issue that hinders programmer productivity. Another contribution of this dissertation is to extend the array dataflow analysis to handle a subset of X10 programs. We apply the result of dataflow analysis to statically guarantee determinism. Providing static guarantees can significantly increase programmer productivity by catching questionable implementations at compile-time, or even while programming

Substructurality and residuation in logic and algebra

A very and natural way of introducing a logic is by using a sequent calculus, or Gentzen system. These systems are determined by specifying a set of axioms and a set of rules. Axioms are then starting points from which we can derive new consequences by using the rules. Hilbert systems consist also on a set of axioms and a set of rules that are used to deduce consequences. The main difference is that, whereas the axioms in Hilbert systems are formulas, and the rules allow to deduce certain formulas from other sets of formulas, in the case of Gentzen systems the axioms are sequents and the rules indicate which sequents can be inferred from other sets of sequents. By a sequent we understand a pair hG, Si, where G and S are finite sequences of formulas. We denote the sequent hG, Si by G . S.1 The sequent G . S intends to formalize – at least in its origin – the concept “the conjunction of all the formulas of G implies the disjunction of all the formulas of S.” The notion of a sequent calculus was invented by G. Gentzen in order to give axiomatizations for Classical and Intuitionistic Propositional Logics. And the rules he gave in both cases can be grouped in different categories: because of its character, the Cut rule deserves a special category for itself; then we have the rules of introduction and elimination of each one of the connectives, both on the left and on the right – of the symbol . –; and finally a set of rules that do not involve any particular connective. These rules are necessary in Classical and Intuitionistic logics because in these logics 1Traditional notations for sequents are G ) S and G ` S, but since both the symbols ) and ` have many other meanings, we prefer to denote sequents by using the less overloaded symbol ., which can also be found in literature with this use. the order in which we are given the premises, or if we have them repeated, is irrelevant, and we do not loose consequences if we extend the set of hypotheses. But there are other logics that do not satisfy all these rules: for instance, relevance logics and linear logic. At first, these logics were studied separately, and different theories were developed for their investigation. But later on, researches arrived to the conclusion that all of them share a common feature, which became more apparent after the work of W. Blok and D. Pigozzi. It was discovered that (pointed) residuated lattices – or FL algebras – are the algebraic counterpart of substructural logics. In the XIX century, Boole noticed a close connection between “the laws of thought,” as he put it, and algebra. After him, other mathematicians put together all the pieces and described a sort of algebras, named Boole algebras after him, and shed light on the connection anticipated by Boole: Boole algebras are the “natural” semantics for Classical Propositional Logic. More connections were discovered between other logics and other sorts of algebras: for instance, Heyting algebras are the “natural” semantics for Intuitionistic Propositional Logic, and MV algebras for Łukasievicz Multivalued Logic. But it was not until 1989, when Blok and Pigozzi published their book Algebraizable Logics, that for the first time the connections between these logics and classes of algebras were finally described with absolute precision. According to their definitions, these classes of algebras are the equivalent algebraic sematics of the corresponding logics. That is, these classes of algebras are the algebraic counterparts of the corresponding logics. Their ideas paved the way to a new branch of mathematics called Abstract Algebraic Logic, which investigates the connections between logics and classes of algebras, and the so-called bridge theorems: that is, theorems that establish bridges between some property of one realm (logic or algebra) with another property of the other realm. The core of the connection between substructural logics and residuated lattices is that in all these logics, some theorem of the following form could always be proven. Thus, we could think that the metalogical symbol ’,’ is acting as a real connective. More precisely, we could introduce a new connective , called fusion, and impose the following rule. Given an algebraic model with a lattice reduct, it is usually the case that the meet and join operations serve as the interpretations of the conjunction and disjunction connectives. What should be then the interpretation of the fusion? Usually, the elements of the lattice are thought as different degrees of truth, and “a . b is provable” is interpreted as “for every assignment, the degree of truth of a is less than that of b.” Under this natural interpretation, the condition (1) becomes: That is, the fusion is interpreted as a residuated operation on the lattice. Being the algebraic semantics of substructural logics and containing many interesting subvarieties such as Heyting algebras, MV algebras, and lattice-ordered groups, to name a few, the variety of residuated lattices is of utmost importance to the studies of Logic and Algebra, hence our interest. In this dissertation we carry out some investigations on different problems concerning residuated lattices. In what follows we give a brief description of the contents and organization of this dissertation. Every chapter – except for the first one, which is devoted to setting the preliminaries – starts with an introduction in which the reader will find a lengthier explanation of the subject of the chapter, the way the material is organized, and references. We start by compiling in Chapter 1 all the essential well-known results about residuated lattices that we will need in the subsequent chapters. We present here the definitions of those concepts that are not specific to some particular chapter, but general. We define the variety of residuated lattices, and some of its more significant subvarieties. We also introduce nuclei, and nucleus retracts. As it is widely known, the lattice of normal convex subalgebras of a residuated lattice is isomorphic to its congruence lattice, and hence its importance. But it turns out that also the lattice of convex (not necessarily normal) subalgebras is of great significance, specially in the case of e-cyclic residuated lattices. Many of its properties depend on the fact that it is a pseudo-complemented lattice. Actually, it is a Heyting algebra. For instance, polars are special sets usually defined in terms of a certain notion of orthogonality; in the case of e-cyclic residuated lattices, polars are the pseudo-complements of the convex subalgebras. We end the chapter by briefly explaining the notions of semilinearity and projectability for residuated lattices. In the 1960’s, P. F. Conrad and other authors set in motion a general program for the investigation of lattice-ordered groups, aimed at elucidating some order-theoretic properties of these algebras by inquiring into the structure of their lattices of convex `-subgroups. This approach can be naturally extended to residuated lattices and their convex subalgebras. We devote Chapters 2 and 3 to two different problems that can be framed within Conrad’s program for residuated lattices. More specifically, in Chapter 2 we revisit the Galatos-Tsinakis categorical equivalence between integral GMV algebras and negative cones of `-groups with a nucleus, showing that it restricts to an equivalence of the full subcategories whose objects are the projectable members of these classes. Afterwards, we introduce the notion of Gödel GMV algebras, which are expansions of projectable integral GMV algebras by a binary term that realizes a positive Gödel implication in every such algebra. We see that Gödel GMV algebras and projectable integral GMV algebras are essentially the same thing. Analogously, Gödel negative cones are those Gödel GMV algebras whose residuated lattice reducts are negative cones of `-groups. Thus, we turn projectable integral GMV algebras and negative cones of projectable `-groups into varieties by including this implication in their signature. We prove that there is an adjunction between the categories whose objects are the members of these varieties and whose morphisms are required to preserve implications. We devote Chapter 3 to the study of certain kinds of completions of semilinear residuated lattices. We can find in the literature different notions of completions for residuated lattices, like for example Dedekind-McNeil completions, regular completions, complete ideal completions, . . . Very often it happens that for a certain algebra in a variety of residuated lattices, those completions exists but do not belong to the same variety. That is, varieties are not closed, in general, under the operations of taking these kinds of completions. But there are other notions of completions that might have better properties in this regard. Conrad and other authors proved the existence of lateral completions, projectable completions, and orthocompletions for representable `-groups, and moreover, that the varieties of representable `-groups are closed under these completions. Our goal in this chapter is to prove the existence of lateral completions, (strongly) projectable completions, and orthocompletions for semilinear e-cyclic residuated lattices, as they are a natural generalization of representable `-groups. We introduce all these concepts along the chapter, and prove first that every semilinear e-cyclic residuated lattice can be densely embedded into another residuated lattice which is latterly complete and strongly projectable. We obtain this lattice as a direct limit of a certain family of algebras obtained from the original lattice by taking quotients and products, so the direct limit stays in the same variety where the original algebra lives. Finally, we prove that for semilinear GMV algebras, we can find minimal dense extensions satisfying all the required properties. In Chapter 4 we study the failure of the Amalgamation Property on several varieties of residuated lattices. The Amalgamation Property is of particular interest in the study of residuated lattices due to its relation with various syntactic interpolation properties of substructural logics. There are no examples to date of non-commutative varieties of residuated lattices that satisfy the Amalgamation Property. The variety of semilinear Abstract 5 residuated lattices is a natural candidate for enjoying this property, since most varieties that have a manageable representation theory and satisfy the Amalgamation Property are semilinear. However, we prove that this is not the case, and in the process we establish that the same happens for the variety of semilinear cancellative residuated lattices, that is, it also lacks the Amalgamation Property. In addition, we prove that the variety whose members have a distributive lattice reduct and satisfy the identity x(y ^ z)w xyw ^ xzw also fails the Amalgamation Property. In Chapter 5 we show how some well-known results of the theory of automata, in particular those related to regular languages, can be viewed within a wider framework. In order to do so, we introduce the concept of module over a residuated lattice, and show that modules over a fixed residuated lattice – that is, partially ordered sets acted upon by a residuated lattice – provide a suitable algebraic framework for extending the concept of a recognizable language as defined by Kleene. More specifically, we introduce the notion of a recognizable element of a residuated lattice by a finite module and provide a characterization of such an element in the spirit of Myhill’s characterization of recognizable languages. Further, we investigate the structure of the set of recognizagle elements of a residuated lattice, and also provide sufficient conditions for a recognizable element to be recognized by a Boolean module. We summarize in Chapter 6 the main results of this dissertation and propose some of the problems that still remain open. We end this dissertation with an appendix on directoids. These structures were introduced independently three times, and their aim is to study directed ordered sets from an algebraic perspective. The structures that we have studied in this dissertations have an underlying order, but moreover they have a lattice reduct. That is not always the case for directed ordered sets. Hence the importance of the study of directoids. We prove some properties of directoids and their expansions by additional and complemented directoids. Among other results, we provide a shorter proof of the direct decomposition theorem for bounded involute directoids. We present a description of central elements of complemented directoids. And finally we show that the variety of directoids, as well as its expansions mentioned above, all have the strong amalgamation property