Monadic Algebras Research Articles

AN ALGEBRAIC PROOF OF COMPLETENESS FOR MONADIC FUZZY PREDICATE LOGIC

Abstract Monoidal t-norm based logic $\mathbf {MTL}$ is the weakest t-norm based residuated fuzzy logic, which is a $[0,1]$ -valued propositional logical system having a t-norm and its residuum as truth function for conjunction and implication. Monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ that consists of the formulas with unary predicates and just one object variable, is the monadic fragment of fuzzy predicate logic $\mathbf {MTL\forall }$ , which is indeed the predicate version of monoidal t-norm based logic $\mathbf {MTL}$ . The main aim of this paper is to give an algebraic proof of the completeness theorem for monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and some of its axiomatic extensions. Firstly, we survey the axiomatic system of monadic algebras for t-norm based residuated fuzzy logic and amend some of them, thus showing that the relationships for these monadic algebras completely inherit those for corresponding algebras. Subsequently, using the equivalence between monadic fuzzy predicate logic $\mathbf {mMTL\forall }$ and S5-like fuzzy modal logic $\mathbf {S5(MTL)}$ , we prove that the variety of monadic MTL-algebras is actually the equivalent algebraic semantics of the logic $\mathbf {mMTL\forall }$ , giving an algebraic proof of the completeness theorem for this logic via functional monadic MTL-algebras. Finally, we further obtain the completeness theorem of some axiomatic extensions for the logic $\mathbf {mMTL\forall }$ , and thus give a major application, namely, proving the strong completeness theorem for monadic fuzzy predicate logic based on involutive monoidal t-norm logic $\mathbf {mIMTL\forall }$ via functional representation of finitely subdirectly irreducible monadic IMTL-algebras.

On saturated prefilter monads

In this paper we show that the prime saturated prefilter monads are sup-dense and interpolating in saturated prefilter monads. It follows that CNS spaces are the lax algebras for prime saturated prefilter monads. As for the algebraic part, we prove that the Eilenberg-Moore algebras for saturated prefilter monads are exactly continuous I-lattices.

State monadic residuated lattices and their corresponding filters

In this paper, we introduce the notion of state monadic residuated lattices and study some of their related properties. Then we prove that the relationship between state monadic algebras of substructural fuzzy logics completely maintains the relationship between corresponding monadic algebras. Moreover, we introduce state monadic filters of state monadic residuated lattice, giving a state monadic filter generated by a nonempty subset of a residuated lattice, and obtain some characterizations of maximal and prime state monadic filters. Finally, we give some characterization of special kinds of state monadic residuated lattices, including simple, semisimple and local state monadic residuated lattices by state monadic filters.

Quantum monadic algebras

We introduce quantum monadic and quantum cylindric algebras. These are adaptations to the quantum setting of the monadic algebras of Halmos, and cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic treatments of classical and intuitionistic predicate logic. Primary examples in the quantum setting come from von Neumann algebras and subfactors. Here we develop the basic properties of these quantum monadic and cylindric algebras and relate them to quantum predicate logic.

Monadic Boolean algebras with an automorphism and their relation to $${\mathbf{Df}}_{\mathbf{2}}$$-algebras

In this work, we initiate an investigation of the class $${\mathcal {B}}_{T_km}$$ of monadic Boolean algebras endowed with a monadic automorphism of period k. These algebras constitute a generalization of monadic symmetric Boolean algebras. We determine the congruences on these algebras and we characterize the subdirectly irreducible algebras. This last result allows us to prove that $${\mathcal {B}}_{T_km}$$ is a discriminator variety and as a consequence, the principal congruences are characterized. Finally, we explore, in the finite case, the relationship between this class and the class $${\mathbf{Df}}_{\mathbf{2}}$$ of diagonal-free two-dimensional cylindric algebras.

Subordination Tarski algebras

ABSTRACTIn this work we will study Tarski algebras endowed with a subordination, called subordination Tarski algebras. We will define the notion of round filters, and we will study the class of irreducible round filters and the maximal round filters, called ends. We will prove that the poset of all round filters is a lattice isomorphic to the lattice of the congruences that are compatible with the subordination. We will prove that every end is an irreducible round filter, and that in a topological subordination Tarski algebra A, every irreducible round filter is an end iff A is a monadic subordination Tarski algebra. As corollary of this result we have that the variety of monadic Tarski algebra can be characterised as the topological Tarski algebras where every irreducible open filter is a maximal open filter.

Shapely monads and analytic functors

In this paper, we give precise mathematical form to the idea of a structure whose data and axioms are faithfully represented by a graphical calculus; some prominent examples are operads, polycategories, properads, and PROPs. Building on the established presentation of such structures as algebras for monads on presheaf categories, we describe a characteristic property of the associated monads---the shapeliness of the title---which says that “any two operations of the same shape agree”. An important part of this work is the study of analytic functors between presheaf categories, which are a common generalisation of Joyal's analytic endofunctors on sets and of the parametric right adjoint functors on presheaf categories introduced by Diers and studied by Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among the analytic endofunctors, and may be characterised as the submonads of a universal analytic monad with “exactly one operation of each shape”. In fact, shapeliness also gives a way to define the data and axioms of a structure directly from its graphical calculus, by generating a free shapely monad on the basic operations of the calculus. In this paper we do this for some of the examples listed above; in future work, we intend to do so for graphical calculi such as Milner's bigraphs, Lafont's interaction nets, or Girard's multiplicative proof nets, thereby obtaining canonical notions of denotational model.

Monadic bounded hoops

The main goal of this paper is to investigate monadic bounded hoops and prove the completeness of the monadic hoop logic. In the paper, we introduce monadic bounded hoops: A variety of bounded hoops equipped with universal and existential quantifiers. Also, we study some properties of them and obtain some conditions under which a bounded hoop becomes a Heyting algebra and BL-algebra. In addition, we introduce and investigate monadic filters in monadic bounded hoops. Using monadic filters on monadic bounded hoops, we characterize simple monadic bounded hoops. Moreover, we focus on algebraic structures of the set $$\textit{MF}[H]$$ of all monadic filters on monadic bounded hoops and obtain that $$\textit{MF}[H]$$ forms a complete Heyting algebra. Further, we discuss relations between monadic bounded hoops and some related algebraic structures, likeness other monadic algebras, bounded hoops with regular Galois connection and rough approximation spaces. Finally, as an application of monadic bounded hoops, we prove the completeness of monadic hoop logic. These results will provide a more general algebraic foundations of soft computing intended as a method for dealing with uncertain information, fuzzy information and decision making.

Complete Elgot Monads and Coalgebraic Resumptions

Monads are used to abstractly model a wide range of computational effects such as nondeterminism, statefulness, and exceptions. Complete Elgot monads are monads that are equipped with a (uniform) iteration operator satisfying a set of natural axioms, which allows to model iterative computations just as abstractly. It has been shown recently that extending complete Elgot monads with free effects (e.g. operations of sending/receiving messages over channels) canonically leads to generalized coalgebraic resumption monads, which were previously used as semantic domains for non-wellfounded guarded processes. In this paper, we continue the study of complete Elgot monads and their relationship with generalized coalgebraic resumption monads. We give a characterization of the Eilenberg-Moore algebras of the latter. In fact, we work more generally with Uustalu's parametrized monads; we introduce complete Elgot algebras for a parametrized monad and we prove that they form an Eilenberg-Moore category. This is further used for establishing a characterization of complete Elgot monads as those monads whose algebras are coherently equipped with the structure of complete Elgot algebras for the parametrized monads obtained from generalized coalgebraic resumption monads.

Algebraic Kan extensions along morphisms of internal algebra classifiers

An “algebraic left Kan extension” is a left Kan extension which interacts well with the algebraic structure present in the given situation, and these appear in various subjects such as the homotopy theory of operads and in the study of conformal field theories. In the most interesting examples, the functor along which we left Kan extend goes between categories that enjoy universal properties which express the meaning of the calculation we are trying to understand. These universal properties say that the categories in question are universal examples of some categorical structure possessing some kind of internal structure, and so fall within the theory of “internal algebra classifiers” described in earlier work of the author. In this article conditions of a monad-theoretic nature are identified which give rise to morphisms between such universal objects, which satisfy the key condition of Guitart-exactness, which guarantees the algebraicness of left Kan extending along them. The resulting setting explains the algebraicness of the left Kan extensions arising in operad theory, for instance from the theory of “Feynman categories” of Kaufmann and Ward, generalisations thereof, and also includes the situations considered by Batanin and Berger in their work on the homotopy theory of algebras of polynomial monads.

Eilenberg–Moore Monoids and Backtracking Monad Transformers

We develop an algebraic underpinning of backtracking monad transformers in the general setting of monoidal categories. As our main technical device, we introduce Eilenberg--Moore monoids, which combine monoids with algebras for strong monads. We show that Eilenberg--Moore monoids coincide with algebras for the list monad transformer ('done right') known from Haskell libraries. From this, we obtain a number of results, including the facts that the list monad transformer is indeed a monad, a transformer, and an instance of the MonadPlus class. Finally, we construct an Eilenberg--Moore monoid of endomorphisms, which, via the codensity monad construction, yields a continuation-based implementation a la Hinze.

Dynamical Systems in Categories

In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We discuss that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, strongly σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

Algebraic functions in quasiprimal algebras

A function is algebraic on an algebra if it can be implicitly defined by a system of equations on . In this note we give a semantic characterization for algebraic functions on quasiprimal algebras. This characterization is applied to obtain necessary and sufficient conditions for a quasiprimal algebra to have every one of its algebraic functions be a term function. We also apply our results to particular algebras such as finite fields and monadic algebras.

Generalized Płonka Sums and Products

We give an abstract categorical treatment of Płonka sums and products using lax and oplax morphisms of monads. Płonka sums with sup-semilattices as arities were originally defined as operations on algebras of regular theories. It turns out that even more general operations are available on the categories of algebras of semi-analytic monads. Their arities are the categories of the regular polynomials over any sup-semilattice, i.e. any algebra for the terminal semi-analytic monad. We also show that similar operations can be defined on the category of algebras of any analytic monad. This time we can allow the arities to be the categories of linear polynomials over any commutative monoid, i.e. any algebra for the terminal analytic monad. There are also dual operations of Płonka products. They can be defined on Kleisli categories of commutative monads.

On independent varieties and some related notions

We investigate the relation of independence between varieties, as well as a generalisation of such which we call strict quasi-independence. Concerning the former notion, we specify a procedure for constructing an independent companion of a given solvable subvariety of a congruence modular variety; we show that joins of independent varieties inherit Mal’cev properties from the joinands; we investigate independence in 3- and 4-permutable varieties; we provide a more economical axiomatisation for the join of two independent varieties than the ones available in the literature. We also explore the latter notion, showing inter alia that joins of strictly quasi-independent varieties inherit the congruence extension property and the strong amalgamation property from the joinands, and conversely. An application section investigates independent varieties of Boolean algebras with operators (in particular, Akishev and Goldblatt’s bounded monadic algebras) and of groups. In particular, a complete characterisation of independent varieties of groups is given.

Martin-Löf complexes

In this paper we define Martin-Löf complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-Löf type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-Löf complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-Löf complexes are a model of homotopy 1-types.

Involutive Categories and Monoids, with a GNS-Correspondence

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive categories.

Monadic Algebras: a Standpoint on Rough Sets

The paper studies complete and incomplete information systems, i.e. basic structures of rough set theory (RST), from the standpoint of monadic Boolean algebras (MBAs). In the first part we recall a simple characterisation of RST as a theory of MBAs. In the second part the paper discusses approximation operators obtained from incomplete information systems. In this case we lose a direct link to MBAs and thus our aim is to obtain an MBA which will additionally be interesting from the perspective of RST. In the third part we consider a simple application of approximation operators based on this MBA.

Kleisli category and database mappings

This paper presents the semantics of database mappings in the relational database (DB) category, based on the power-view monad T and monadic algebras. The semantics can be interpreted as a computational model of view-based mappings between databases, where each query (view-mapping) can be seen as a program, so that we can use the paradigm 'from values to computations'. The objects in this category are the database-instances. The morphisms of such DB category are used in order to express the semantics of view-based global and local as view (GLAV) mappings between relational databases such as those used in data integration systems. Consequently, the semantics of database mappings in this DB category are defined based on the power-view monad T and the Kleisli category for databases, which can be 'internalised' in this basic DB category.

On monadic quantale algebras: basic properties and representation theorems

Motivated by the concept of quantifier (in the sense of P. Halmos) on different algebraic structures (Boolean algebras, Heyting algebras, MV-algebras, orthomodular lattices, bounded distributive lattices) and the resulting notion of monadic algebra, the paper introduces the concept of a monadic quantale algebra, considers its properties and provides several representation theorems for the new structures.