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

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

Geometric, Algebraic, and Topological Combinatorics

The 2019 Oberwolfach meeting "Geometric, Algebraic and Topological Combinatorics" was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) Karim Adiprasito presented his very recent proof of the $g$-conjecture for spheres (as a talk and as a "Q\&A" evening session) (2) Federico Ardila gave an overview on "The geometry of matroids", including his recent extension with Denham and Huh of previous work of Adiprasito, Huh and Katz

Triangulations

The earliest work in topology was often based on explicit combinatorial models – usually triangulations – for the spaces being studied. Although algebraic methods in topology gradually replaced combinatorial ones in the mid-1900s, the emergence of computers later revitalized the study of triangulations. By now there are several distinct mathematical communities actively doing work on different aspects of triangulations. The goal of this workshop was to bring the researchers from these various communities together to stimulate interaction and to benefit from the exchange of ideas and methods

The Resemblance Structure of Natural Kinds: A Formal Model for Resemblance Nominalism

278 p.The aim of this thesis is to better understand the ways natural kinds are related to each other by species-genus relations and the ways in which the members of the kind are related to each other by resemblance relations, by making use of formal models of kinds. This is done by first analysing a Minimal Conception of Natural Kinds and then reconstructing it from the ontological assumptions of Resemblance Nominalism. The questions addressed are:(1) What is the external structure of kinds' In what ways are kinds related to each other by species-genus relations'(2) What is the internal structure of kinds' In what sense are the instances of a kind similar enough to each other'According to the Minimal Conception of Kinds, kinds have two components, a set of members of the kind (the extension) and a set of natural attributes common to these objects (the intension). Several interesting features of this conception are discussed by making use of the mathematical theory of concept lattices. First, such structures provide a model for contemporary formulations of syllogistic logic. Second, kinds are ordered forming a complete lattice that follows Kant's law of the duality between extension and intension, according to which the extension of a kind is inversely related to its intension. Finally, kinds are shown to have Aristotelian definitions in terms of genera and specific differences. Overall this results in a description of the specificity relations of kinds as an algebraic calculus.According to Resemblance Nominalism, attributes or properties are classes of similar objects. Such an approach faces Goodman's companionship and imperfect community problems. In order to deal with these, a specific nominalism, namely Aristocratic Resemblance Nominalism, is chosen. According to it, attributes are classes of objects resembling a given paradigm. A model for it is introduced by making use of the mathematical theory of similarity structures and of some results on the topic of quasianalysis. Two other models (the polar model and an order-theoretic model) are considered and shown to be equivalent to the previous one.The main result is that the class of lattices of kinds that a nominalist can recover uniquely by starting from these assumptions is that of complete coatomistic lattices. Several other related results are obtained, including a generalization of the similarity model that allows for paradigms with several properties and properties with several paradigms. The conclusion is that, under nominalist assumptions, the internal structure of kinds is fixed by paradigmatic objects and the external structure of kinds is that of a coatomistic lattice that satisfies the Minimal Conception of Kinds