global asymptotic and exponential stability of a dynamic neural system with asymmetric connection weights recently a dynamic neural system was presented and analyzed due to its good performance in optimization computation and low complexity for implementation the global asymptotic stability of such a dynamic neural system with symmetric connection weights was well studied in this note based on a new lyapunov function we investigate the global asymptotic stability of the dynamic neural system with asymmetric connection weights since the dynamic neural system with asymmetric weights is more general than that with symmetric ones the new results are significant in both theory and applications specially the new result can cover the asymptotic stability results of linear systems as special cases

enriched categories and many valued preorders a question from programming arises if bitstring x compares with bitstring y to some degree and if bitstring y satisfies predicate a to some degree then how should the possibility be mathematically modeled that bitstring x satisfies predicate a to at least some degree related to both and mathematically modeling this question is surprisingly intricate when the underlying conjunctions e g of predicates are non commutative potential applications of this question occur in data mining a field in which pattern matching is important and commonly used this paper uses ideas from enriched categories over monoidal categories to address such issues and enable pattern matching techniques to extend to many valued contexts equipped with non commutative conjunctions first we consider the notion of a set enriched by a po monoid l which turns out to be a set equipped with an l valued preorder second we extend the notion of enriched functors to formulate the appropriate definition of variable basis morphisms between preordered sets over different lattice theoretic bases third we construct the notion of topological systems enriched with many valued preorders and use their extent spaces to motivate and formulate enriched or preordered topological spaces the compatibility or enrichment axioms for such systems and spaces model the programming question stated above using non commutative tensor products en route are determined a number of related notions and results many valued antisymmetry characterizes the l t0 separation axiom many valued preorders are categorically topological and many valued partial orders are monotopological enriched topological spaces form a topological category a large inventory of example classes is provided including programming examples and an extensive discussion of examples based on the l spectra of complete po groupoids and each of these related developments is provided a suitable lattice theoretic and categorical foundation

adaptive observers with arbitrary exponential rate of convergence for nonlinear systems concerns the same class of linearly parameterized single output nonlinear systems that the authors previously identified in 1992 in terms of differential geometric conditions when persistency of excitation conditions are satisfied the adaptive observers presented in this note guarantee arbitrarily fast exponential convergence both of parameter and state estimates to actual parameters and states while previous adaptive observers guarantee only exponential not arbitrarily fast convergence this extends earlier results for linear systems

connecting fuzzy preorders fuzzy consequence operators and fuzzy closure and co closure systems abstract the relations between fuzzy preorders fuzzy consequence operators fuzzy closure systems and co closure systems lattice structures are studied given a continuous t norm we will prove that the complete lattice of all fuzzy preorders is isomorphic with a subfamily of the complete lattice of fuzzy consequence operators moreover we will show that it is possible to embed both structures in the class of fuzzy closure systems by a lattice dual monomorphism between semilattices a solution to the following open problem when two families of fuzzy possible valuations produce the same fuzzy implication relation is given concept of interior operator of a fuzzy consequence operator is suggested and a characterization of the previous open problem through interior operator is shown

fixed points of adjoint functors enriched in a quantaloid abstract representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid in a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis fca and rough set theory rst which not only extend the realm of formal contexts to multi typed and multi valued ones but also provide a general approach to construct various kinds of representation theorems besides incorporating several well known representation theorems in fca and rst as well as formulating new ones it is shown that concept lattices in rst can always be represented as those in fca through relative pseudo complements of the given contexts especially if the contexts are valued in a non girard quantaloid

fuzzy sets and sheaves part ii this essay shows that large parts of fuzzy set theory are actually subfields of sheaf theory resp of the theory of complete w valued sets hence fuzzy set theory is closer to the mainstream in mathematics than many people would expect part ii of this series of two papers explains the sheaf theoretic basis of zadeh s operations on fuzzy sets with the exception of the complementation further the quotient problem w r t similarity relations and the quotient problem of fuzzy groups w r t fuzzy congruence relations are solved finally stratified fuzzy topologies are identified with internal topologies on constant sheaves

some corollaries of the correspondence between partial metrics and multivalued equalities multivalued equalities originally arose in the context of the theory of sheaves while partial metrics originally arose in the context of the theory of domains for denotational semantics it turns out that multivalued equalities and partial metrics are closely related one can either consider them equivalent up to the choice of dual notation or one can formally capture the duality between logical and metric viewpoints by explicitly requiring logical values and distances to be represented by dual structures not only does this allow the transfer of the ideas and results between these two fields but the most interesting situations are those of interplay between logical and metric considerations the computational understanding of partial metrics as upper bounds and of multivalued equalities as lower bounds is discussed together with the lower bound counterparts to partial metrics and the corresponding upper bound counterparts to multivalued equalities it is shown that separated pre sheaves of sets and functions over complete heyting algebras can be understood as the pre sheaves of ultrametrics valued in the corresponding brouwerian algebras and non expansive maps of those ultrametrics it is proposed that the natural logical counterparts for partial metrics valued in non negative reals are multivalued equalities valued in the quantale of non positive reals the issue of canonical partial metrics on higher order and reflexive scott domains is revisited the intuition behind strong triangularity vickers form the computational version of strong triangularity and weighted quasi metrics are discussed in the context of quantaloid enrichment

categorical foundations of topology with applications to quantaloid enriched topological spaces based on ordered monads this paper uncovers the categorical basis of topology in terms of a categorical formulation of neighborhood axioms here dense subobjects lower separation axioms and regularity receive a purely categorical representation in the case of appropriate submonads of the double presheaf monad this theory is applied to quantaloid enriched topological spaces which form a common framework for many valued topology as well as for non commutative topology as an illumination of this situation two examples are given the first one is chosen from probability theory and has the following characteristics weak convergence of smooth probability measures is topological the hausdorff separation axiom is valid dirac measures form a dense subset the second example is related to operator theory and explains the topologization of spectra of non commutative c algebras

categorical approaches to non commutative fuzzy logic in this paper we consider what it means for a logic to be non commutative how to generate examples of structures with a non commutative operation which have enough nice properties to serve as the truth values for a logic inference in the propositional logic is gotten from the categorical properties products coproducts monoidal and closed structures adjoint functors of the categories of truth values we then show how to extend this view of propositional logic to a predicate logic using categories of propositions about a type a with functors giving change of type and adjoints giving quantifiers in the case where the semantics takes place in set l goguen s category of l fuzzy sets the categories of predicates about a can be represented as internal category objects with the quantifiers as internal functors

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

fuzzy presubsets as non idempotent and non commutative classifications of subalgebras in the framework of quantale preordered sets regular q fuzzy presubsets are identified with classifiable subalgebras this identification is based on the monad t of quasi presingletons which is an extension of the monad associated with the cauchy completion of quantale preordered sets even though not every t subalgebra is classifiable the classifiable hull of t subalgebras exists always an application of these constructions is the internalization of the negation of the underlying quantale as truth arrow in the category of t algebras

lattice valued preordered sets as lattice valued topological systems this paper provides variable basis lattice valued analogues of the well known results that the construct prost of preordered sets firstly is concretely isomorphic to a full concretely coreflective subcategory of the category top of topological spaces which employs the concept of the dual of the specialization preorder and secondly is non concretely isomorphic to a full coreflective subcategory of the category topsys of topological systems of s vickers which employs the spatialization procedure for topological systems dualizing these results one arrives at the similar properties of quasi pseudo metric spaces built over locales

roughness and fuzziness in quantales abstract rough set theory and fuzzy set theory are two useful mathematical tools for dealing with uncertainty and granularity in information systems motivated by the studies of roughness and fuzziness in algebraic systems and partially ordered sets such as semigroups rings and lattices in this paper we introduce first the notions of fuzzy prime semi prime primary ideals of quantales and investigate their properties several characterizations of such ideals are presented then we introduce the concepts of weak fuzzy prime ideals and strong fuzzy prime ideals of quantales and establish relationships between these ideals and fuzzy prime ideals of quantales by applying rough set theory to fuzzy ideals of quantales we furthermore define rough fuzzy prime semi prime primary ideals of quantales generalizing yang and xu s work on quantales to the fuzzy environment finally relationships between the upper resp lower rough fuzzy ideals of quantales and the upper resp lower approximations of their homomorphic images are also discussed

semitopological groups semiclosure semigroups and quantales abstract as a valued domain quantales have been applied to the study of enriched category fuzzy algebra fuzzy topology and fuzzy domain one purpose of this paper is to investigate semitopological groups and semiclosure semigroups from the viewpoint of quantales we first show that a semitopological group is completely determined by the topological ideal conuclei of the power set quantale over a group next we prove that the concepts of semitopological groups and semiclosure groups are dually equivalent another purpose of this paper is to investigate ordered semigroups conditionally complete quantales and quantales from the viewpoint of semiclosure semigroups as an important result of semiclosure semigroups we prove that the category of complete semiclosure semigroups is isomorphic to the category of quantales

preordered sets valued in a gl monoid let m l be a gl monoid an m valued preordered set is an l subset endowed with a reflexive and m transitive l relation it is essentially a category enriched in a quantaloid generated by m this paper presents a study of m valued preordered sets with emphasis on symmetrization and the cauchy completion the main result states that symmetrization and the cauchy completion of m valued preordered sets commute up to a natural isomorphism

numerical representation of transitive fuzzy relations we show that there exists a continuous fuzzy quasi order r on the set of real numbers r such that for any fuzzy quasiorder r on a finite set x there is a mapping f x r such that r x y r f x f y for all x y x

free q algebras abstract the main purpose of this paper is to consider some methods for constructing q algebras and to give the concrete form of a free unital q algebra over a set a semigroup an ordered semigroup

neural network for quadratic optimization with bound constraints a recurrent neural network is presented which performs quadratic optimization subject to bound constraints on each of the optimization variables the network is shown to be globally convergent and conditions on the quadratic problem and the network parameters are established under which exponential asymptotic stability is achieved through suitable choice of the network parameters the system of differential equations governing the network activations is preconditioned in order to reduce its sensitivity to noise and to roundoff errors the optimization method employed by the neural network is shown to fall into the general class of gradient methods for constrained nonlinear optimization and in contrast with penalty function methods is guaranteed to yield only feasible solutions

fuzzy submonoids fuzzy preorders and quasi metrics let l be a residuated lattice then given a monoid m we define a galois connection from the lattice of the compatible l preorders in m and the lattice of l submonoids of m given a set s we define a galois connection between the lattice of the l preorders in s and the lattice of l submonoids of the monoid ss i a link with the notion of quasi metric is also established

the double power monad is the composite power monad in the context of quantaloid enriched categories we rely essentially on the classifying property of presheaf categories to give a conceptual proof of a theorem due to hhle the double power monad and the composite power monad on the category of quantaloid enriched categories are the same via the theory of distributive laws we identify the algebras of this monad to be the completely codistributive complete categories and the homomorphisms between such algebras are the bicontinuous functors with these results we hope to contribute to the further development of a theory of q valued preorders in the sense of pu and zhang

fuzzy galois connections on fuzzy sets abstract in fairly elementary terms this paper presents how the theory of preordered fuzzy sets more precisely quantale valued preorders on quantale valued fuzzy sets is established under the guidance of enriched category theory motivated by several key results from the theory of quantaloid enriched categories this paper develops all needed ingredients purely in order theoretic languages for the readership of fuzzy set theorists with particular attention paid to fuzzy galois connections between preordered fuzzy sets

simple involutive quantales abstract involutive quantales were introduced in 7 as complete lattices equipped with a multiplication and an involution such structures are well known from the calculus of relations the set r el x of binary relations on any setxforms an involutive quantale however the motivating example of an involutive quantale is the spectrum max aof a non commutativec algebraa where max ais the involutive quantale consisting of all closed linear subspaces ofa in 8 it was indeed shown thatacan be reconstructed from the involutive quantale max a demonstrating that involutive quantales do provide a convenient algebraic invariant of non commutativec algebras the aim of this paper is to study involutive quantales from the algebraic point of view our main result is the characterization of simple involutive quantales that is involutive quantales on which the only surjections are isomorphisms or constant morphisms we also show that this class contains all quantales q s of sup preserving endomorphisms on a complete latticeshaving a duality a fortiori all hilbert quantales as defined in 7 a special case of this is r el x which can be seen to be q 2x our investigation of the class of simple involutive quantales is also motivated byc algebras it is proved in 8 that for anyc algebraa max ahas enough homomorphisms into the hilbert quantales q p h where p h is the complete lattice of closed linear subspaces of a hilbert spaceh in this spirit viewing involutive quantales as duals of generalized non commutative topological spaces simple involutive quantales correspond to spaces having only trivial subspaces that is to points we begin a study of those involutive quantales which have enough homomorphisms into simple involutive quantales noting their spatial character such a study has algebraic antecedents in the early work 4 and the more recent work 1 which studied certain lattices noncomplete involutive quantales having enough homomorphisms into simple involutive quantales of the form r el x

variety theorem for algebras with fuzzy orders abstract we present generalization of the bloom variety theorem of ordered algebras in fuzzy setting algebras with fuzzy orders consist of sets of functions which are compatible with fuzzy orders fuzzy orders are defined on universe sets of algebras using complete residuated lattices as structures of degrees in this setting we show that classes of models of fuzzy sets of inequalities are closed under suitably defined formations of subalgebras homomorphic images and direct products conversely we prove that classes having these closure properties are definable by fuzzy sets of inequalities

development and analysis of a neural dynamical approach to nonlinear programming problems this technical note develops a neural dynamical approach to nonlinear programming np problems whose equilibrium points coincide with karush kuhn tucker points of the np problem a rigorous analysis on the global convergence and the convergence rate of the proposed neural dynamical approach is carried out under the condition that the associated lagrangian function is convex analysis results show that the proposed neural dynamical approach can solve general convex programming problems and a class of nonconvex programming problems two nonconvex programming examples are provided to demonstrate the performance of the developed neural dynamical approach

fuzzy preorder and fuzzy topology this paper presents by means of alexandrov topology for fuzzy preordered sets and specialization order for fuzzy topological spaces a systematic investigation of the interrelationship between fuzzy preordered sets topological spaces and fuzzy topological spaces

fuzzy terms in this paper we will show how purely categorical constructions of terms are advantageous when investigating situations concerning uncertainty more specifically where uncertainty comes from and how uncertainty is integrated when dealing with terms over selected signatures there are basically two ways of invoking uncertainty for terms on one hand we may proceed by building composed monads where uncertainty is provided by some suitable monad composed with the traditional term monad on the other hand we can provide a strictly formal basis for term monads being created over categories themselves carrying uncertainty this is the distinction between computing with fuzzy and fuzzy computing and the fundamental question raised by these constructions is where uncertainty resides in language constructions for logic this paper also shows how the notion of signature often needs to be expanded to levels of signatures in particular when dealing with type constructors such levels allow us to strictly delineate e g primitive operations type terms and value level terms levels of signature will in this paper be exemplified by the construction of the signature of simply typed lambda calculus

a formal study of linearity axioms for fuzzy orderings abstract this contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings different existing concepts are evaluated with respect to three fundamental correspondences from the classical case linearizability of partial orderings intersection representation and one to one correspondence between linearity and maximality as a main result we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case if we do not require a one to one correspondence between linearity and maximality however we obtain that an implication based definition appears to constitute a sound compromise in particular if ukasiewicz type logics are considered

an analysis of a class of neural networks for solving linear programming problems a class of neural networks that solve linear programming problems is analyzed the neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact nondifferentiable penalty functions it is proved that for a given linear programming problem and sufficiently large penalty parameters any trajectory of the neural network converges in finite time to its solution set for the analysis lyapunov type theorems are developed for finite time convergence of nonsmooth sliding mode dynamic systems to invariant sets the results are illustrated via numerical simulation examples

on the injective hulls of quantum b algebras abstract quantum b algebras as implicational subreducts of quantales were introduced by rump and yang which cover the majority of implicational algebras and provide a unified semantics for non commutative algebraic logic due to the adjunction between multiplication and its residuals they can be regarded as the implicational counterpart of partially ordered semigroups by means of the injective hulls envelopes of quantum b algebras rump gave the injective hulls of partially ordered semigroups in this paper one part of our work is to give new examples of quantum b algebras where the lower sets do not form a quantale another part of our work is to consider the inverse problem of rump s way that is we can also give the injective hulls of quantum b algebras in terms of the injective hulls of partially ordered semigroups

a novel neural network for a class of convex quadratic minimax problems based on the inherent properties of convex quadratic minimax problems this article presents a new neural network model for a class of convex quadratic minimax problems we show that the new model is stable in the sense of lyapunov and will converge to an exact saddle point in finite time by defining a proper convex energy function furthermore global exponential stability of the new model is shown under mild conditions compared with the existing neural networks for the convex quadratic minimax problem the proposed neural network has finite time convergence a simpler structure and lower complexity thus the proposed neural network is more suitable for parallel implementation by using simple hardware units the validity and transient behavior of the proposed neural network are illustrated by some simulation results

sierpinski object for affine systems abstract motivated by the concept of sierpinski object for topological systems of s vickers presented recently by r noor and a k srivastava this paper introduces the sierpinski object for many valued topological systems and shows that it has three important properties of the crisp sierpinski space of general topology

diagonals between q distributors abstract for a small quantaloid q it is shown that the category of q distributors and diagonals is equivalent to a quotient category of the category of q interior spaces and continuous q distributors kan adjunctions induced by q distributors play a crucial role in establishing this equivalence

quantale valued dissimilarity abstract inspired by the theory of apartness relations of scott we establish a positive theory of dissimilarity valued in an involutive quantale q without the aid of negation it is demonstrated that a set equipped with a q valued dissimilarity is precisely a symmetric category enriched in a subquantaloid of the quantaloid of back diagonals of q interactions between q valued dissimilarities and q valued similarities which are equivalent to q valued equalities in the sense of hohle kubiak are investigated with the help of lax functors in particular it is shown that similarities and dissimilarities are interdefinable if q is a girard quantale with a hermitian and cyclic dualizing element

on convergence rate of projection neural networks this note presents an analysis of the convergence rate for a projection neural network with application to constrained optimization and related problems it is shown that the state trajectory of the projection neural network is exponentially convergent to its equilibrium point if the jacobian matrix of the nonlinear mapping is positive definite while the convergence rate is proportional to a design parameter if the jacobian matrix is only positive semidefinite moreover the convergence time is guaranteed to be finite if the design parameter is chosen to be sufficiently large furthermore if a diagonal block of the jacobian matrix is positive definite then the corresponding partial state trajectory of the projection neural network is also exponentially convergent three optimization examples are used to show the convergence performance of the projection neural network

quantale valued preorders each divisible and unital quantale is associated with two quantaloids categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders this paper deals with the relationship between these two kinds of enriched categories this is a special case of the change base issue in enriched category theory two functors the forward globalization functor and the backward globalization functor that map a fuzzy set with a fuzzy preorder to a crisp set with a fuzzy preorder are considered it is shown that both of them preserve cocompleteness with respect to several classes of weights

a non commutative and non idempotent theory of quantale sets in fuzzy set theory non idempotency arises when the conjunction is interpreted by arbitrary t norms there are many instances in mathematics where set theory ought to be non commutative and or non idempotent the purpose of this paper is to combine both ideas and to present a theory of non commutative and non idempotent quantale sets among other things standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases more specifically the category q set is investigated where q is an arbitrary involutive quantale objects quantale sets are pairs consisting of a set and a q valued equality with a suitable symmetry axiom three important properties of q set are shown it is complete cocomplete and has the epi extremal mono factorization property its subcategory s q set of separated quantale sets is reflective in q set and shares the same fundamental properties with q set in particular s q set is also a complete and cocomplete epi extremal mono category the objects of the category q set are interesting categories in their own right two categorical frameworks for objects of q set are exhibited first it is shown that q valued equalities arise from q valued preorders with self adjoint extents by symmetrization which leaves q valued equalities invariant here sets with quantale preorders are shown to be b categories with base b being a specific quantaloid the second approach is based on involutive quantaloids a combination of two well known things quantaloids and ordered categories with involution in this context quantale sets are precisely symmetric b categories w r t appropriately chosen quantaloid b with involution further the cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames our non commutative generalization of frames and all left continuous t norms there exist at least two monads on q set the singleton monad and quasi singleton monad which are of special interest for fuzzy set theory the kleisli category associated with the singleton monad is the non commutative and non idempotent analogue of higgs topos while the eilenberg moore category of the quasi singleton monad permits the internalization of lukasiewicz negation as truth arrow finally an application of quantale sets to c algebras is given and the change of base is treated

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

stratified categorical fixed basis fuzzy topological spaces and their duality abstract for an abstract category c a class m of c monomorphisms and a fixed c object l we introduce stratified c m l spaces to be categorical counterparts of stratified fixed basis fuzzy topological spaces in c and consider their category sc m l top as two main results of this paper it is shown that sc m l top is dually adjoint to the comma category l c and this adjunction can be restricted to a dual equivalence between the full category of l c with comma spatial objects and the full category of sc m l top with comma sober objects the present paper also describes applications and relationships of these results to stratified fixed basis fuzzy topological spaces in this respect a considerable part of this paper is devoted to stratified l quasi topological spaces and their duality

generalized henon maps the cubic diffeomorphisms of the plane abstract in general a polynomial diffeomorphism of the plane can be transformed into a composition of generalized henon maps these maps exhibit some of the familiar properties of the quadratic henon map including a bounded set of bounded orbits and an anti integrable limit we investigate in particular the cubic area preserving case which reduces to two two parameter families of maps the bifurcations of low period orbits of these maps are discussed in detail

q closure spaces for a small quantaloid q a q closure space is a small category enriched in q equipped with a closure operator on its presheaf category we investigate q closure spaces systematically with specific attention paid to their morphisms and as preordered fuzzy sets are a special kind of quantaloid enriched categories in particular fuzzy closure spaces on fuzzy sets are introduced as an example by constructing continuous relations that naturally generalize continuous maps it is shown in the generality of the q version that the category of closure spaces and closed continuous relations is equivalent to the category of complete lattices and sup preserving maps

towards probabilistic partial metric spaces diagonals between distance distributions abstract the quantale of distance distributions is of fundamental importance for understanding probabilistic metric spaces as enriched categories motivated by the categorical interpretation of partial metric spaces we are led to investigate the quantaloid of diagonals between distance distributions which is expected to establish the categorical foundation of probabilistic partial metric spaces observing that the quantale of distance distributions w r t an arbitrary continuous t norm is non divisible we precisely characterize diagonals between distance distributions and prove that one step functions are the only distance distributions on which the set of diagonals coincides with the generated down set

foundations of fuzzy sets abstract this paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice it then gives detailed accounts of categorical approaches using a closed structure to capture the fuzzy and connective within these categories weak forms of subobject representations provide an internal second order logic an approach to fuzzy real numbers and fuzzy topology is included to illustrate the use of this internal second order theory