short communication some remarks on the category fuz h of m eytan the aim of this paper is to disprove a result asserted by m eytan in his paper fuzzy sets a topos logical point of view in this journal vol 5 pp 47 67 namely that the category fuz h is a topos for any complete heyting algebra h it is proved that this result can be true only when h is a boolean algebra

h fuzzy relations i a topological universe viewpoint abstract we introduce the subcategory rel r h of rel h consisting of fuzzy reflexive relational space on sets and we study structures of rel r h in a viewpoint of the topological universe introduced by l d nel we show that rel r h is a topological universe over set also we introduce the subcategories rel pr h rel p h rel e h and rel po h of rel r h we will show that rel pr h resp rel p h and rel e h is a topological universe over set

the set valued mapping based on ample fields ample fields play an important role in possibility theory based on the ample fields in possibility theory two new concepts such as possibilistic set and possibility mapping are presented a theoretical approach to define operations and implication operations of fuzzy sets based on the theory of falling shadows of possibilistic sets is established and the category posm of ample spaces and possibility mappings and the category x w a of possibilistic sets are built up respectively it is proved that the category x w a is a topos

short communication the categories set h and fuz h this paper follows up d ponasse s paper some remarks on the category fuz h of m eytan in this journal 4 in particular it gives a new proof that fuz h cannot have a subobject classifier if h is a complete chain with at least three elements

a survey of fuzzy set and topos theory abstract this paper is a comparison and contrast of approaches to many valued mathematics offered by fuzzy set theory and topos theory it gives a survey of the categorical foundations of fuzzy set theory and related topoi topoi are not a basis for fuzzy set theory butthey do suggest appropriate directions to go and questions to ask for a synthesis which does provide a foundation one possible structure which has a lopos like internal logic and the rich variety of loggical connectives used in fuzzy sets is included

a partial information non zero sum differential game of backward stochastic differential equations with applications this paper is concerned with a new kind of non zero sum differential game of backward stochastic differential equations bsdes it is required that the control is adapted to a sub filtration of the filtration generated by the underlying brownian motion we establish a necessary condition in the form of maximum principle with pontryagin s type for open loop nash equilibrium point of this type of partial information game and then give a verification theorem which is a sufficient condition for nash equilibrium point the theoretical results are applied to study a partial information linear quadratic lq game and a partial information financial problem

on a categorical analysis of zadeh generalized subsets of sets ii how can fuzzy sets be viewed as generalized subsets of actual sets an answer to this question is given via a categorical analysis of the synchronic approach of fuzzy set theory starting from an abstract definition of a category with fuzzy subsets we establish necessary and sufficient conditions on a posst to be that of truth values for such a category these conditions justify zadeh s original choice of the unit segment as a set of truth values some defects concerning universal constructions in a non trivial category with fuzzy subsets are mentioned and a natural best toposophical approximation of such a category is proposed

brief paper maximum principle for the stochastic optimal control problem with delay and application in this paper we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay we obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations our results can be applied to a production and consumption choice problem the explicit optimal consumption rate is obtained

reconstruction possibilities for fuzzy sets towards a new level of understanding abstract using the methodological approach of rational reconstruction some possible explications of the notion of fuzzy set are critically discussed with respect to their intuitive meaning plausibility and consistency together with a historical survey we try to provide a better understanding of the development of various concepts within traditional fuzzy set theory based on the approaches of 39 and 18 we show that zadeh s original intuition of a fuzzy set and its formalization as generalized membership function can be reinterpreted in a plausible and consistent way and thus are supported not only on a formal level by mathematical results but can also be justified in principle on a semantical level hence both the operational as well as the traditional valuational viewpoint have their own rights in their corresponding areas of modeling reality and it is shown how the latter can be given a sound foundation based upon the former

fuzzy sets a topos logical point of view abstract benabou deduction categories are defined with a set of additional assumptions that define categories with formal finite limits resp formal regular categories formal logoi formal topoi they are shown to be generalized structures in which higher order many sorted languages can be realized the corresponding gentzen type higher order calculus of sequents is explicited and the soundness theorem is formulated a construction is given which associates to each deduction category with formal properties a real category with the corresponding real properties in a universal way the corresponding sounddess and completeness properties are formulated for the real categories thus obtained fuzzy sets as generalized by goguen are introduced considered as the objects of a category fuz h which turns out to be the real category associated to a very simple formal topos and thus to be itself a topos furthermore this is proved to be a grothendieck topos which is a strictly full epireflective subcategory of higgs category of h valued sets topoi are proposed as generalized fuzzy sets and deductio0 categories as generalized 2 fuzzy sets some related topics such as arbib manes fuzzy theories probability many valued and fuzzy logics intensional logic are very briefly touched upon

maximum principle for nonzero sum stochastic differential game with delays in this technical note we discuss a nonzero sum stochastic differential game with delays not only the state variable but also control variables of players involve delays this kind of games are motivated by some interesting problems arising from economics and finance using anticipated backward stochastic differential equations we establish a necessary condition and a sufficient condition of maximum principle for the delayed game problem to explain theoretical results we apply them to an economic problem

fuzzy sets do not form a topos abstract the relationship between fuzzy sets and h valued sets is briefly examined and a claim of eytan s that the former comprise the objects of a topos is corrected

categorical analysis of logic operators on fuzzy sets abstract although the concept of topos can lead naturally to the logic operations of the class of sets seth cannot be connected to the logic operations of fuzzy sets due to the fact that sell i has no subobject classifier in thsi work from category fuz we introduce several concepts such as middle object wsc and wtopos it is shown that by the use of category fuz which is a wtopos the fuzzy logic operations can be obtained both logically and naturally

fuzzy sets and toposes towards higher order fuzzy logic abstract a series of paper which have appeared in fuzzy sets and systems over the last three years are reviewed in an expository manner the claims and counter claims made in them are dealt with some of the problems of representing knowledge in the first order predicate calculus are discussed briefly and a case is made for exposing the relationship between topoi and fuzzy mathematics with a view to making a true fuzzy logic both consistent complete and respectable the paper is an attempt to present the controversy to the non specialist and especially to practitioners in expert systems and artificial intelligence

smooth partitions of unity on banach spaces abstract we show a new characterisation of the existence of smooth partitions of unity on a banach space this leads to a slight generalisation of some as well as a very easy recovery of most of the known results using a unified treatment

completion of l fuzzy relations it is shown that the category l of complete l similarities on l sets is a full reflective subcategory of r l l fuzzy graphs l is equivalent to lh l sheaves on l connections with other known fuzzy categories are also studied

categorical properties of f set theory abstract in the first part of the paper we show that the interpretation of a fuzzy set as a bundle of sets over a base space makes the category of fuzzy sets into a topos f set v i e to paraphrase pitts 5 fuzzy sets do form a topos the second part exhibit close connections to operations proposed for fuzzy set theory from an intuitive and operational viewpoint by toth 7 these two results together seem to indicate that there are two essentially different ways to understand the concept of a fuzzy set either as an evaluation over a universe or as a collection of evaluated objects the latter one being a set theoretical interpretation and henceforth leading to a topos

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

real values multisets and fuzzy sets abstract we develop a first order theory mstr for positive real valued multisets by restricting multiplicities of elements to the real interval 0 1 we obtain a theory mstf that gives a zermelo fraenkel like axiomatization for zadeh s fuzzy sets moreover mstr and mstf contain an exact copy of classical zfc set theory and both are shown to be relatively consistent mstf is compared to other axiom systems that have been proposed for fuzzy sets

categories of fuzzy sets and weak topos in this paper definitions of middle object in a category and concept of weak topos are given a weak topos is a cartesian closed category with a middle object it is proved that category of functors from a small category to category fuz of fuzzy sets is a weak topos

a pontryagin s maximum principle for non zero sum differential games of bsdes with applications this technical note is concerned with a maximum principle for a new class of non zero sum stochastic differential games the most distinguishing feature compared with the existing literature is that the game systems are described by backward stochastic differential equations bsdes this kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities we establish a necessary condition and a sufficient condition in the form of maximum principle for open loop equilibrium point of the foregoing games respectively to explain the theoretical results we use them to study a financial problem

mathematical aspects of fuzzy sets and fuzzy logic the paper considers some of the main trends of the development of mathematical fuzzy logic and of formalized fuzzy set theory which came out in the history of fuzzy sets and which aim to give solid theoretical foundations for fuzzy set theory

smoothness in weakly compactly generated banach spaces received august 25 1982 revised april 15 1983 if x is a weakly compactly generated wcg banach space then x admits an equivalent c smooth norm if a wcg banach space x admits a ck smooth function with bounded support then x admits ck smooth partitions of unity

categorial studies of fuzzy sets abstract this paper presents some recent studies on the categorial aspects of fuzzy sets and sets with a valued equality in order to suitably develop a theory fuzzy sets it seems useful to construct a category such an approach is very helpful when a classical concept has to be generalized for performing a choice between all the apparently possible definitions such a category has besides to look as much as possible like the usual category of sets one can wish it would be a topos or at least something similar

algebraic structures in fuzzy logic abstract common features in various fuzzy logical systems are investigated it turns out that adjoint couples and residuated lattices are very often present though not always explicitly expressed some minor new results are also proved

the stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls this paper is concerned with a pontryagin s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls this kind of control problems is motivated by some interesting phenomena arising from economics and finance we establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis to explain the theoretical results we apply them to a production and consumption choice problem

anticipated backward stochastic differential equations in this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations anticipated bsdes in these equations the generator includes not only the values of solutions of the present but also the future we show that these anticipated bsdes have unique solutions a comparison theorem for their solutions and a duality between them and stochastic differential delay equations

a new category for fuzzy topological spaces abstract a new category for fuzzy topological spaces is defined that includes some important categories proposed so far a new category for fuzzy sets is also proposed

nonzero sum stochastic differential games with discontinuous feedback the existence of a nash equilibrium feedback is established for a two player nonzero sum stochastic differential game with discontinuous feedback this is obtained by studying a parabolic system strongly coupled by discontinuous terms

when does a category built on a lattice with a monoidal structure have a monoidal structure in a word sometimes and it gets harder if the structure on l is not commutative in this paper we consider the question of what properties are needed on the lattice l equipped with an operation for several different kinds of categories built using sets and l to have monoidal and monoidal closed structures this works best for the goguen category set l in which membership but not equality is made fuzzy and maps respect membership commutativity becomes critical if we make the equality fuzzy as well this can be done several ways so a progression of categories is considered using sets with an l valued equality and functions which respect that equality gives a monoidal category which is closed if we use a strong form of the transitive law if we use strict extensional total relations and a strong transitive law and is commutative and nearly idempotent we get a monoidal structure we also recall some constructions by mulvey nawaz and hohle on quantales with properties making them commutative enough to have non symmetric monoidal structures

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

categoryfuz wtopos and categoryfuzfuz abstract a new category fuzfuz is introduced from category fuz of fuzzy sets and its topoi properties are investigated although category fuzfuz is not a topos it has middle object and wsc and consequently fuzfuz can also form a wtopos just as category fuz we also prove that category fuzfuz has power objects

topoi and categories of fuzzy sets abstract let h be a completely distributive lattice and hence a heyting algebra goguen s category of fuzzy sets set h eytan s logos fuz h and the topos of sheaves on h sh h are interconnected by pairs of adjoint functors between them each of the categories has a predicate calculus these predicate calculi are related through the functors between the categories change of base lattice gives rise to several functors which preserve or reflect specific kinds of statements in the predicate calculi this paper gives details of the categories the predicate calculi attached to them the functors between the categories and the preservation properties of those functors