on the maslov index in this paper we give four definitions of maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other moreover relationships of several symplectic and differential geometric analytic and topological invariants including triple maslov indices eta invariants spectral flow and signatures of quadratic forms to the maslov index are developed and formulae relating them are given the broad presentation is designed with a view to applications both in geometry and in analysis 1994 john wiley sons inc

implication structures fuzzy subsets and enriched categories we show that there are natural ways to make a fuzzy set into an enriched category and that in doing so some basic results in fuzzy set theory are applications of enriched category theory thus we argue that the theory of enriched categories is a useful tool for fuzzy set theorists

a comparative study of ideals in fuzzy orders abstract this paper presents a comparative study of three kinds of ideals in quantale valued order theory forward cauchy ideals generated by forward cauchy nets flat ideals and irreducible ideals main results include necessary and sufficient conditions for forward cauchy ideals to be flat or irreducible for integral quantales

a non frame valued cartesian closed category of liminf complete fuzzy orders let h 0 12 1 with the natural order and p q max p q1 0 for all p qh it is proved that the category of liminf complete h ordered sets is cartesian closed it reveals that there exists a cartesian closed category consisting of non frame valued liminf complete fuzzy ordered sets

closedness of the category of liminf complete fuzzy orders it is shown that the category of liminf complete fuzzy orders and liminf continuous maps valued in the complete residuated lattice based on a continuous t norm on the unit interval is monoidal closed and that this category is cartesian closed if and only if the t norm is the minimum

dual morse index estimates for periodic solutions of hamiltonian systems in some nonconvex superquadratic cases in l ekeland and hofer stated a very general result for the existence of periodic solutions with any prescribed minimal period for hamiltonian systems assuming that the hamiltonian function h is strictly convex with superquadratic behaviour at the origin and at infinity the arguments given in l are based on an estimate for the morse index of a critical point of mountain pass type for a functional which is associated with the hamiltonian system via the duality principle of clarke and ekeland 2 unfortunately the functional is only c and not c then the result is achieved only through a finite dimensional reduction and as the authors themselves say through an inordinate amount of technicalities in this paper we give a direct definition of the morse index of a periodic solution which does not use the duality principle indeed it is based on a study of the morse indexes of approximating solutions obtained by a galerkin scheme such as the one introduced by rabinowitz in 3 taking into account some results about linking point theory recently stated in 4 71 we get the same index estimates as in l but in a simple way moreover some generalizations are obtained in the sense that the convexity assumption on h can be replaced either by a local convexity condition or by a condition on the second derivative of h plus eventually a symmetry condition see also 8 91 for some other results obtained in this framework in particular in the locally convex case the index estimates yield direct information on the minimal period of the corresponding solution finally let us remark that this definition of the morse index can be adapted to other nonconvex cases and one can still get some index estimates of the same type

homoclinic solutions of an infinite dimensional hamiltonian system we consider the system begin equation left begin array l partial t u delta x u v x u h v t x u v 3mm partial t v delta x v v x v h u t x u v end array qquad text for t x in mathbb r times mathbb r n right end equation which is an unbounded hamiltonian system in l 2 mathbb r n mathbb r 2m we assume that the constant function u 0 v 0 equiv 0 0 in mathbb r 2m is a stationary solution and that h and v are periodic in the t and x variables we present a variational formulation in order to obtain homoclinic solutions z u v satisfying z t x to 0 as t x to infty it is allowed that v changes sign and that delta v has essential spectrum below and above 0 we also treat the case of a bounded domain omega instead of mathbb r n with dirichlet boundary conditions

fundamental study complete and directed complete ω categories let w be a commutative unital quantale complete and directed complete w categories are the core objects in quantitative domain theory this paper based on the theory of f completeness for enriched categories presents a systematic investigation of completeness and directed completeness of w categories

partial quasi metrics in this article we introduce and investigate the concept of a partial quasi metric and some of its applications we show that many important constructions studied in matthews s theory of partial metrics can still be used successfully in this more general setting in particular we consider the bicompletion of the quasi metric space that is associated with a partial quasi metric space and study its applications in groups and bck algebras

fuzzifying interval operators fuzzifying convex structures and fuzzy pre orders abstract the paper presents essential connections between fuzzifying convexities fuzzifying interval operators and fuzzy pre orders firstly we study the relationship between fuzzifying convex structures and fuzzifying interval operators on a universal set secondly we show that there exists a connection between based point fuzzy orders and geometric fuzzifying interval operators on fuzzy partially ordered sets finally we establish a connection between fuzzifying convexities and fuzzy pre orders

liminf convergence in ω categories abstract we investigate and unify notions of convergence in preorders and metric spaces the basic structures we use are categories enriched over commutative unital quantales such structures encompass both preorders and metric spaces as was observed already by lawvere 1974 in this report we define a notion of convergence on the general structures that generalizes least upper bounds of chains in preorders and limits of cauchy sequences in metric spaces thereby the notions of complete partial order and complete metric space are also unified as are the two versions of continuous functions we use this unified setting to indicate how recursive domain equations can be treated by providing a generalization of scott s inverse limit theorem the theorem as well as its proof specializes to scott s original ones for the preorder case and to america and rutten s for the metric case we also give a categorical analysis of our notion of convergence comparing it to weighted limits and colimits in enriched categories

the l ordered semigroups based on l partial orders abstract fuzzy frame valued ordered semigroups are investigated with the orders being l partial orders in this framework l left resp l right ideals l bi ideals and l quasi ideals of an l ordered semigroup are discussed the idea of which is applied to the characterizations of regular and duo l ordered semigroups in addition the relationship between an l ordered semigroup and an l topological semigroup is studied as a consequence an isomorphism between the category of l ordered semigroups and the category of l topological semigroups with the topologies being l t 0 and alexandrov is obtained

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

a computational model for metric spaces for every metric space x we define a continuous poset bx such that x is homeomorphic to the set of maximal elements of bx with the relative scott topology the poset bx is a dcpo iff x is complete and continuous iff x is separable the computational model bx is used to give domain theoretic proofs of banach s fixed point theorem and of two classical results of hutchinson on a complete metric space every hyperbolic iterated function system has a unique non empty compact attractor and every iterated function system with probabilities has a unique invariant measure with bounded support we also show that the probabilistic power domain of bx provides an continuous computational model for measure theory on a separable complete metric space x

automorphisms of the canonical anticommutation relations and index theory let h be a complex hilbert space p an orthogonal projection on h and p the complementary projection if g is any symmetrically normed ideal in the ring of bounded operators on h then we consider the group of unitary operators on h such that p up and p u p lie in g when g is the hilbert schmidt class these unitaries define automorphisms of the c algebra b of the canonical anticommutation relations over h which are implementable in the representation of b determined by p we investigate the structure of the group u proving in particular that it has infinitely many connected components uk labelled by the fredholm index of p u p the connected component of the identity u0 is generated by unitaries of the form exp ia with a self adjoint and p a p in g finally we consider an application of these results to two dimensional field theory showing in particular that the charge and chiral charge quantum numbers arise as the fredholm indices of p up for certain unitary u on l2 r c2

the existence of periodic solution for infinite dimensional hamiltonian systems abstract in this paper we will consider a kind of infinite dimensional hamiltonian system hs by the method of saddle point reduction topology degree and the index defined in wang and liu 2015 we will get the existence of periodic solution for hs

dynamical systems measures and fractals via domain theory abstract we introduce domain theory in dynamical systems iterated function systems fractals and measure theory for a discrete dynamical system given by the action of a continuous map f x x on a metric space x we study the extended dynamical systems vx vf ux uf and lx lf where v u and l are respectively the vietoris hyperspace the upper hyperspace and the lower hyperspace functors we show that if x f is chaotic then so is ux uf when x is locally compact ux is a continuous bounded complete dcpo if x is second countable as well then ux will be continuous and can be given an effective structure we show how strange attractors attractors of iterated function systems fractals and julia sets are obtained effectively as fixed points of deterministic functions on ux or fixed points of non deterministic functions on cux where c is the convex plotkin power domain we also show that the set m x of finite borel measures on x can be embedded in pux where p is the probabilistic power domain this provides an effective framework for measure theory we then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on pux

ideal models of spaces ideal domains have an elementary order theoretic structure every element is either compact or maximal despite this we establish that 1 they can model any space currently known to possess a countably based model and 2 the metric spaces with ideal models are exactly the completely metrizable spaces

effective domain representations of h x the space of compact subsets this paper gives effective domain representations of spaces h x of non empty compact subsets of effective complete metric spaces x the domain representation of h x is constructed from a domain representation of x using the plotkin power domain construction as an application of the representation an effective version of a fundamental theorem on ifs iterated function system is shown

two cartesian closed subcategories of fuzzy domains the aim of this paper is to search for cartesian closed subcategories of fuzzy domains based on the notion of retraction embedding pair in the category of fuzzy dcpos we prove that the category of fuzzy continuous lattices and the category of fuzzy algebraic lattices are cartesian closed

static space times naturally lead to quasi pseudometrics the standard 4 dimensional minkowski space time of special relativity is based on the 3 dimensional euclidean metric in 1967 h busemann showed that similar static space time models can be based on an arbitrary metric space in this paper we search for the broadest possible generalization of a metric under which a construction of a static space time leads to a physically reasonable space time model it turns out that this broadest possible generalization is related to the known notion of a quasi pseudometric

yoneda completeness and flat completeness of ordered fuzzy sets abstract this paper studies yoneda completeness and flat completeness of ordered fuzzy sets valued in the quantale obtained by endowing the unit interval with a continuous triangular norm both of these notions are natural extension of directed completeness in order theory to the fuzzy setting yoneda completeness requires every forward cauchy net converges has a yoneda limit while flat completeness requires every flat weight a counterpart of ideals in partially ordered sets has a supremum it is proved that flat completeness implies yoneda completeness but the converse implication holds only in the case that the related triangular norm is either isomorphic to the ukasiewicz t norm or to the product t norm

a superversion of quasifree second quantization i charged particles a formalism comprising and extending quasifree second quantization of charged bosons and fermions is presented the second quantization of one particle observables leads to current superalgebras and a super schwinger term shows up anticommuting parameters are introduced in order to construct super bogoliubov transformations mixing bosons and fermions as an application representations of lie superalgebras are given which are semidirect products of extensions of affine kac moody algebras and the virasoro algebra and of the super virasoro algebra

std convergence in fuzzy metric spaces in this note we answer two recent questions posed by morillas and sapena 10 related to standard convergence in fuzzy metric spaces in the sense of george and veeramani the obtained results lead us to establish what conditions must satisfy a concept about sequential convergence to be considered compatible with a concept of cauchyness

complete relations on fuzzy complete lattices abstract we generalize the notion of complete binary relation on complete lattice to residuated lattice valued ordered sets and show its properties then we focus on complete fuzzy tolerances on fuzzy complete lattices and prove they are in one to one correspondence with extensive isotone galois connections finally we prove that any fuzzy complete lattice factorized by a complete fuzzy tolerance is again a fuzzy complete lattice

twisted condensates of quantised fields the authors construct some quasi free pure states of three quantised fields in 1 1 dimensions that are localised in the sense of knight 1961 they consider massless or massive dirac fields forming a u n n or 1 multiplet and subject it to a local gauge transformation they also subject a doublet of massive klein gordon fields to local so 2 transformations they find the conditions that the resulting automorphisms are spatial in fock space in some cases the conditions turn out to require that certain parameters identified as the winding numbers of the gauge are integers it is argued that this integer labels states of various charge

a relative morse index for the symplectic action the notion of a morse index of a function on a finite dimensional manifold cannot be generalized directly to the symplectic action function a on the loop space of a manifold in this paper we define for any pair of critical points of a a relative morse index which corresponds to the difference of the two morse indices in finite dimensions it is based on the spectral flow of the hessian of a and can be identified with a topological invariant recently defined by viterbo and with the dimension of the space of trajectories between the two critical points

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

quantitative domains via fuzzy sets part i continuity of fuzzy directed complete posets this paper deals with quantitative domain theory via fuzzy sets it examines the continuity of fuzzy directed complete posets dcpos for short based on complete residuated lattices first we show that a fuzzy partial order in the sense of fan and zhang and an l order in the sense of belohlavek are equivalent to each other then we redefine the concepts of fuzzy directed subsets and continuous fuzzy dcpos we also define and study fuzzy galois connections on fuzzy posets we investigate some properties of continuous fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy double downward arrow operator has a right adjoint we define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy way below relation is the smallest approximating fuzzy auxiliary relation

cartesian closedness of a category of non frame valued complete fuzzy orders abstract let h 0 1 2 1 with the natural order and p q max p q 1 0 for all p q h we know that the category of liminf complete h ordered sets is cartesian closed in this paper it is proved that the category of conically cocomplete h ordered sets with liminf continuous functions as morphisms is cartesian closed more importantly a counterexample is given which shows that the function spaces consisting of liminf continuous functions of complete h ordered sets need not be complete thus the category of complete h ordered sets with liminf continuous functions as morphisms is not cartesian closed

an application of a representation theorem for fuzzy metrics to domain theory in this paper we study the interplay between fuzzy metric spaces and domain theory using the representation theorem established in mardones perez and de prada vicente 2012 12 we will prove that suitable restrictions of completeness in fuzzy metric spaces correspond to a structure of domain in the poset of formal balls endowed with appropriate orders

continuity spaces reconciling domains and metric spaces abstract we use continuity spaces a common refinement of posets and metric spaces to develop a general theory of semantic domains which includes metric spaces and domains of cpo s as special cases and provides the appropriate tools for producing new examples which may be suitable for modeling language constructs that occur in concurrent and probabilistic programming our proposal for a general notion of semantic domain is a symmetrically compact continuity space where is a value quantale we show that the category of symmetrically compact continuity spaces with continuous maps has many of the key properties required of a category of domains and that it captures in a natural way the traditional examples in general the category will not be cartesian closed however powerdomains do exist and by adapting a construction of suenderhauf to continuity spaces we show that they define a computational monad in the sense of moggi

the essence of ideal completion in quantitative form if a poset lacks joins of directed subsets one can pass to its ideal completion but doing this means also changing the setting the universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with scott continuous functions as morphisms the same applies for the quantitative version of ideal completion suggested in the literature as in the case of posets it seems advantageous to consider a different topology with the completed spaces we introduce topological v continuity spaces and their smyth completion and show that this is an adequate setting to consider ideal completion of quantitative domains performing the smyth completion of a v continuity space regarded as topological v continuity space gives the ideal completion of the original space together with its scott topology

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

on the yoneda completion of a quasi metric space several theories aimed at reconciling the partial order and the metric space approaches to domain theory have been presented in the literature e g flagg and kopperman theoret comput sci 177 1 1997 111 138 bonsangue et al theoret comput sci 193 1998 1 51 symth quasi uniformities reconciling domains with metric spaces lectures notes in computer science vol 298 springer berlin 1987 pp 236 253 wagner ph d thesis carnegie mellon university pittsburgh july 1994 technical report cmu cs 94 159 we focus in this paper on two of these approaches the yoneda completion of generalized metric spaces of bonsangue et al theoret comput sci 193 1998 1 51 which finds its roots in work by lawvere lawvere rend sem mat fis milano 43 1973 135 166 cf also wagner ph d thesis carnegie mellon university pittsburgh july 1994 technical report cmu cs 94 159 and which is related to early work by stoltenberg e g stoltenberg proc london math soc 3 17 1967 226 240 stoltenberg proc amer math soc 18 1967 864 867 and ferrer and gregori proc london math soc 3 49 1984 36 and the smyth completion smyth quasi uniformities reconciling domains with metric spaces lecture notes in computer science vol 298 springer berlin 1987 pp 236 253 smyth in g m reed a w roscoe r f wachter eds topology and category theory in computer science oxford university press oxford 1991 pp 207 229 smyth j london math soc 49 1994 385 400 snderhauf in m droste y gurevich eds semantics of programming languages and model theory algebra logic and applications vol 5 gordon and breach london 1993 pp 189 212 snderhauf acta math hungar 69 1995 47 54 a net version of the yoneda completion complementing the net version of the smyth completion snderhauf acta math hungar 69 1995 47 54 is given and a comparison between the two types of completion is presented the following open question is raised in bonsangue et al theoret comput sci 193 1998 1 51 an interesting question is to characterize the family of generalized metric spaces for which the yoneda completion is idempotent it contains at least all ordinary metric spaces we show that the largest class of quasi metric spaces idempotent under the yoneda completion is precisely the class of smyth completabe spaces a similar result has been obtained independently by flagg and snderhauf in flagg and snderhauf preprint available at ftp theory doc ic ac uk theory papers sunderhauf eicqf ps 2 we present an entirely new proof of the result via concrete standard techniques and compare this approach with the more abstract categorical machinery of flagg and snderhauf preprint available at ftp theory doc ic ac uk theory papers sunderhauf eicqf ps our proof is based on a new characterization of smyth completability of quasi metric spaces in terms of sequences which considerably simplifies prior characterizations for quasi uniform spaces e g snderhauf in m droste y gurevich eds semantics of programming languages and model theory algebra logic and applications vol 5 gordon and breach london 1993 pp 189 212 sunderhauf acta math hungar 69 1995 715 720 we also show that the ideal completion and hence the yoneda completion and the smyth completion are not sequentially adequate in general the study of the properties of total boundedness precompactness hereditary precompactness and compactness is motivated and we analyze the preservation of these properties under the two kinds of completion in the possible absence of idempotency

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

embedding metric spaces into cpo s abstract we show that metric spaces with polish topology mp can be represented very well by bounded complete algebraic cpo s with continuous weight and distance cwd an operator cwd mp is defined such that 1 and 2 hold 1 1 d can be embedded topologically into d such that a partial function on d is metrically continuous if it can be extended to a total continuous function on d 2 2 every m mp is isometric to some d therefore the continuous cpo functions are exactly adequate for describing the continuous functions on the corresponding metric spaces the approach could be refined in order to study computability on metric spaces with polish topology examples of application are the euclidean space the space of total functions on the natural numbers baire space and spaces of infinite trees

join completions of l ordered sets in this paper we build and characterize the join completions of an l ordered set which generalizes the join completions of a crisp poset the universal property of join completions is studied a categorical characterization of the dedekind macneille completion for an l ordered set is presented

a new index theory for linear self adjoint operator equations and its applications abstract we develop an index theory for linear self adjoint operator equation with the linear self adjoint operator having no compact resolvent as applications we consider the existence and multiplicity of periodic solutions of wave equations and beam equations

domain representations of partial functions with applications to spatial objects and constructive volume geometry a partial spatial object is a partial map from space to data data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry cvg where space and data are subspaces of n dimensional euclidean space we investigate the computability of partial spatial object data types in general and in volume graphics using the theory of effective domain representations for topological algebras the basic mathematical problem considered is to classify which partial functions between topological spaces can be represented by total continuous functions between given domain representations of the spaces we prove theorems about partial functions on regular hausdorff spaces and their domain representations and apply the results to partial spatial objects and cvg algebras

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

elements of generalized ultrametric domain theory generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces as was observed by lawvere 1973 guided by his enriched categorical view on ultra metric spaces we generalize the standard notions of cauchy sequence and limit in an ultra metric space and of adjoint pair between preorders this leads to a solution method for recursive domain equations that combines and extends the standard order theoretic smyth and plotkin 1982 and metric america and rutten 1989 approaches

some properties of m fuzzifying convexities induced by m orders abstract in this paper the notions of geometric interval spaces convex geometries and base point orders in the theory of convexity spaces are generalized to m fuzzifying convexity spaces using m fuzzifying restricted hull operators the m fuzzifying convexity spaces induced by m orders are defined conveniently then it is shown that m fuzzifying convexity spaces induced by m orders are m fuzzifying jhc and arity 2 and if the m order is strongly anti symmetric then the m fuzzifying convexity space is an m fuzzifying convex geometry and its segment operator is an m fuzzifying geometric interval operator

a categorical isomorphism between injective stratified fuzzy t_ bm 0 spaces and fuzzy continuous lattices for a frame l as the truth value table we apply fuzzy domain theory for the study of injective objects in the category of stratified l t 0 spaces we show that every fuzzy continuous lattice equipped with the fuzzy scott topology is an injective stratified l t 0 space and conversely the specialization l ordered set of an injective stratified l t 0 space is a fuzzy continuous lattice these two transformations form a categorical isomorphism between injective stratified l t 0 spaces and fuzzy continuous lattices

the relative morse index theory for infinite dimensional hamiltonian systems with applications abstract we established an index theory for a kind of infinite dimensional linear hamiltonian system as applications we considered the existence and multiplicity of periodic solution for some nonlinear infinite dimensional hamiltonian systems

an enriched category approach to many valued topology in this paper we discuss a basic question in fuzzy set theory what is a many valued topology on a set x an answer is presented by interpreting the axioms two logical formulas for classical topological spaces in the fuzzy setting making use of the intrinsic fuzzy partial order on the fuzzy power set of x as preparation a systematic investigation of fuzzy partial orders is presented from the viewpoint of enriched category theory

relationships between l ordered convergence structures and strong l topologies by making use of the intrinsic fuzzy inclusion orders on the fuzzy power set the relationships between l ordered convergence spaces and strong l topological spaces are researched where l is a commutative unital quantale it is shown that the category of strong l topological spaces can be embedded in the category of l ordered convergence spaces as a reflective subcategory as a result we observe that there is a galois correspondence between the category of l ordered convergence spaces and the category of strong l topological spaces further it is proved that the class of all strong l topological l ordered convergence spaces precisely is the class of all strong l topological spaces and the class of spaces with non idempotent l ordered interior operators is characterized as a subclass of the class of l ordered convergence spaces

computable banach spaces via domain theory this paper extends the order theoretic approach to computable analysis via continuous domains to complete metric spaces and banach spaces we employ the domain of formal balls to define a computability theory for complete metric spaces for banach spaces the domain specialises to the domain of closed balls ordered by reversed inclusion we characterise computable linear operators as those which map computable sequences to computable sequences and are effectively bounded we show that the domain theoretic computability theory is equivalent to the well established approach by pour el and richards

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

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

fermion currents in 1 1 dimensions let f denote the fock space for a free massless fermion field in two space time dimensions if q and q5 denote the charge and chiral charge operators then f n1 n2fn1 n2 where fn1 n2 is the joint eigenspace of q and q5 with corresponding eigenvalues n1 and n2 the smeared time zero free fermion currents formally given by j f x x f x dx 0 1 f s r are in fact self adjoint densely defined operators on f and their exponentials generate a c algebra which leaves each subspace fn1 n2 invariant if a periodic box cutoff for the fermions is introduced uhlenbrock has shown that this c algebra acts irreducibly in each sector fn1 n2 we prove this same result without any cutoffs and determine the properties of the representations in each sector the importance of this result for the problem of constructing fermion fields from observables in two dimensional models such as those of thirring and schwinger is discussed

on fuzzy monotone convergence q cotopological spaces abstract in this paper we generalize the concept of a monotone convergence space also called a d space to the setting of a q cotopological space where q is a commutative and integral quantale we establish a d completion for every stratified q cotopological space which is a category reflection of the category s q ctop of stratified q cotopological spaces onto the full subcategory s q dctop of monotone convergence q cotopological spaces by introducing the notion of a tapered set a direct characterization of the completion is obtained the d completion of each stratified q cotopological space x consists exactly of those tapered closed sets in x we show that the d completion can be applied to obtain a universal fuzzy directed completion of a q ordered set by endowing it with the scott cotopology taking the d completion and then passing to the specialization q order consequently the category q dord of fuzzy directed complete q ordered sets and scott continuous functions is reflective in the category q ord of q ordered sets and scott continuous functions

a survey of fuzzifications of frames the papert papert isbell adjunction and sobriety this paper gives a survey of fuzzifications of frames the papert papert isbell adjunction between the categories of topological spaces and locales as well as the related concept of sobriety it then establishes categorical isomorphisms between yao l frames zhang liu l frames and l algebras lastly it studies the relationships between l sobriety modified l sobriety and i l sobriety the obtained results suggest that modified l sobriety is the most fruitful notion between the above concepts of lattice valued sobriety

some covariant representations of massless fermi fields studies the class of pure quasi free gauge invariant states of the fermion algebra the authors show that the question of g covariance of the corresponding representations i e implementability of some group g of one particle symmetry transformations is related to a cohomology group with values in the hilbert schmidt class in four dimensional space time this cohomology is nontrivial thus leading to g covariant non fock representations when g is taken to be the space time translation group together with rotations or space time translations together with boosts in one direction and rotations about it and the fermion mass is zero in two space time dimensions some new fully poincare covariant representations for free massless fermions are constructed

quantitative domains via fuzzy sets part ii fuzzy scott topology on fuzzy directed complete posets this paper pursues an investigation on quantitative domains via fuzzy sets initiated by the first author this time we define an l topology called the fuzzy scott topology on fuzzy dcpos and investigate its properties scott convergence of stratified l filters is also defined and studied we show that a fuzzy dcpo x e is continuous if and only if for any stratified l filter on x scott convergence coincides with the convergence with respect to the fuzzy scott topology at last we show that the category of fuzzy dcpos with fuzzy scott continuous maps is cartesian closed

total objects of domains abstract the main result of this paper is a characterization of the space induced by scott topology on the set of total objects of a bounded complete cpo this characterization is simple with bounded complete algebraic cpo s where a family of clopen sets plays an essential role in the characterization in the more general case of continuity one can only produce distinct families of open sets and closed sets to axiomatize the essential properties of a total space besides the main characterization we also discuss its relation to continuous spaces studied in our earlier paper and the relation between compact t spaces and total spaces under the lower 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

periodic solutions of an infinite dimensional hamiltonian system abstract we consider the existence of periodic solutions of the unbounded hamiltonian system h t u x u h v t x u v t v x v h u t x u v for t x r unlike previous work in our case the energy functional does not satisfy the palais smale conditions

implementability of gauge transformations and quantization of fermions in external fields quantization of fermions in an external soliton field leading to a representation of the canonical anticommutation relation car which is inequivalent to the representation connected to the massive dirac operator is studied classes of gauge and axial gauge transformations that can be unitarily implemented are determined in the latter case quantization conditions for gauge functions are obtained integers entering can be interpreted as winding numbers

operators on pavelka s algebras induced by fuzzy relations abstract the theory of operators generated by binary fuzzy relations is quickly growing due its fundamental nature and potential applicability the main goal of the paper is to present several representation theorems for operators induced by fuzzy relations on pavelka s algebras for example closure operators used in formal concept analysis monadic operators or tense operators consequently we establish algebraic models with their semantics which are usable in the non classical logic research and in the computer science research

a note on the boson fermion correspondence and infinite dimensional groups we show how to construct irreducible projective representations of the infinite dimensional lie group map s1 mathbb t by embedding it into the group of bogoliubov automorphisms of the car using techniques of g segal for extending certain representations of map s1 su 2 we show that our representations extend to give representations of a certain infinite dimensional superalgebra we relate our work to the well known boson fermion correspondence which exists in 1 1 dimensions

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

generalized metric spaces completion topology and power domains via the yoneda embedding abstract generalized metric spaces are a common generalization of preorders and ordinary metric spaces lawvere 1973 combining lawvere s 1973 enriched categorical and smyth s 1988 1991 topological view on generalized metric spaces it is shown how to construct 1 1 completion 2 2 two topologies and 3 3 powerdomains for generalized metric spaces restricted to the special cases of preorders and ordinary metric spaces these constructions yield respectively 1 1 chain completion and cauchy completion 2 2 the alexandroff and the scott topology and the e ball topology 3 3 lower upper and convex powerdomains and the hyperspace of compact subsets all constructions are formulated in terms of a metric version of the yoneda 1954 embedding

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

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

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

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

a characterization of partial metrizability a characterization of partial metrizability is given which provides a partial solution to an open problem stated by kunzi in the survey paper non symmetric topology in proceedings of the szekszard conference bolyai soc math studies vol 4 1993 pp 303 338 problem 711 characterize those quasi uniformities having a countable base which are induced by a weighted quasi metric the characterization yields a powerful tool which establishes a correspondence between partial metrics and special types of valuations referred to as q valuations cf also theoret comput sci to appear the notion of a q valuation essentially combines the well known notion of a valuation with a weaker version of the notion of a quasi unimorphism i e an isomorphism in the context of quasi uniform spaces as an application we show that continuous directed complete partial orders dcpos are quantifiable in the sense of o neill in s andima et al eds proceedings of the 11th summer conference on general topology and applications annals of the new york academy of sciences vol 86 1997 pp 304 315 i e the scott topology and partial order are induced by a partial metric for algebraic dcpos the lawson topology is induced by the associated metric the partial metrization of general domains improves prior approaches in two ways the partial metric is guaranteed to capture the scott topology as opposed to e g smyth quasi uniformities reconciling domains with metric spaces lecture notes in computer science vol 298 springer berlin 1987 pp 236 253 bonsangue et al theoret comput sci 193 1998 1 flagg theoret comput sci to appear and flagg theoret comput sci 177 1 1997 1 which in general yield a coarser topology partial metric spaces are smyth completable and hence their smyth completion reduces to the standard bicompletion this type of simplification is advocated in smyth in g m reed a w roscoe r f wachter eds topology and category theory in computer science oxford university press oxford 1991 pp 207 229 our results extend smyth 1991 s scope of application from the context of 2 3 sfp domains to general domains the quantification of general domains solves an open problem on the partial metrizability of domains22pawel waszkiewicz communicated recently to the author that he obtained similar results independently a stronger result implying the quantifiability of general domains is reported in 23 o neill obtained the partial metrizability of algebraic domains in his thesis 29 stated in o neil 1997 and heckmann appl categor struct 1999 71 our proof of the quantifiability of domains is novel in that it relies on the central notion of a semivaluation schellekens the correspondence between partial metrics and semivaluations theoret comput sci to appear the characterization of partial metrizability is entirely new and sheds light on the deeper connections between partial metrics and valuations commented on in bukatin and shorina in m nivat ed foundations of software science and computation structures lecture notes in computer science vol 1378 springer berlin 1998 pp 125 139 based on schellekens the correspondence between partial metrics and semivaluations theoret comput sci to appear and our present characterization we conclude that the notion of a semi valuation is central in the context of quantitative domain theory since it can be shown to underlie the various models arising in the applications

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

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

a computable approach to measure and integration theory we introduce a computable framework for lebesgue s measure and integration theory in the spirit of domain theory for an effectively given second countable locally compact hausdorff space and an effectively given finite borel measure on the space we define a recursive measurable set which extends the corresponding notion due to s anin for the lebesgue measure on the real line we also introduce the stronger notion of a computable measurable set where a measurable set is approximated from inside and outside by sequences of closed and open subsets respectively the more refined property of computable measurable sets give rise to the idea of partial measurable subsets which naturally form a domain for measurable subsets we then introduce interval valued measurable functions and develop the notion of recursive and computable measurable functions using interval valued simple functions this leads us to the interval versions of the main results in classical measure theory the lebesgue integral is shown to be a continuous operator on the domain of interval valued measurable functions and the interval valued lebesgue integral provides a computable framework for integration

sobriety of quantale valued cotopological spaces abstract for each commutative and integral quantale making use of the fuzzy order between closed sets a theory of sobriety for quantale valued cotopological spaces is established based on irreducible closed sets