particle lattices and fuzzy particle lattices abstract this paper introduces several new concepts a particle in a lattice which is a generalization of an atom in a distributive lattice by weakening the minimality property of atom a particle lattice which is a generalization of distributive complete atomic lattices by replacing atoms by particles a fuzzy particle lattice a particle lattice with an order reversing involution a particulate boolean algebra a fuzzy particle lattice whose involution is a complementation operator and other related concepts points fuzzy points boundary points and boundary quasi points the paper establishes representation theorems for fomse of these concepts establishes connections between others and some well established lattice theoretic concepts and provides some elementary properties for some of the new concepts

q convergence of nets and weak separation axioms in fuzzy lattices abstract in this paper we introduce the concept of q remote neighborhood and establish the q convergence theory of nets in fuzzy lattices next we introduce the st i i 1 0 1 2 separation axioms in fuzzy lattices and discuss some of their characterizations finally we introduce and study s irresolute order homomorphism and s homeomorphism discuss the application of q convergence of nets and show that the st i i 1 0 1 2 separability is preserved under s homeomorphism

semi global observer for multi output nonlinear systems abstract we present an explicit form of nonlinear observers for a class of multi input multi output systems observer construction for multi output nonlinear systems is not a trivial extension of single output case especially when the global error convergence is of interest in this paper we consider a class of systems in which the subsystem for each output has a triangular dependence on the states of that subsystem itself and the overall system has a block triangular form for each subsystem hence the contribution is to extend the results existing in the literature in that interconnections between the subsystems are allowed the construction is based on the saturation of some estimates which is originated by khalil and esfandiari for the use of semi global output feedback control

on two weakers forms of continuous order homomorphisms the aim of this paper is to find further connections between two different current researches on fuzzy topology the one based on the concepts of neighborhood filter and zadeh function and the other based on the concepts of remote neighborhood ideal and order homomorphism on fuzzy lattices to approach this the paper studies two weaker forms of continuous order homomorphisms the semi continuous order homomorphism and the weakly semi continuous order homomorphism between l fuzzy topological spaces these order homomorphisms can be characterized in each of the notions remote neighborhood neighborhood ideal filter and molecular net in addition some relations connecting weakly semi continuous order homomorphisms semi hausdorff l fts and urysohn l fts have also been established

structures of fuzzy order homomorphisms abstract first the concept of fuzzy order homomorphism is introduced which is a variety of the generalized order homomorphism given in 7 then we show that the fuzzy order homomorphism possesses a nicely local structure furthermore we establish a necessary and sufficient condition under which the fuzz function is a zadeh type function the present condition is simpler than that given in 6 finally for each fuzzy order homomorphism f we yield some characterizations about f 1 1 f which sharpen the corresponding theorem in 6

fuzzy smooth preproximity spaces abstract the concept of a fuzzy smooth preproximity space is introduced and studied also some particular constructions of fuzzy smooth preproximity are given a natural link is established between this concept and the concept of fuzzy pretopological spaces and fuzzy preuniform spaces

the uniform boundedness principle in l topological vector spaces in this paper we introduce and study the equicontinuity of a family of l fuzzy linear order homomorphisms we prove a generalization of the uniform boundedness principle in the classical topological vector spaces to l topological vector spaces

integral action in output feedback for multi input multi output nonlinear systems we address a particular problem of output regulation for multi input multi output nonlinear systems specifically we are interested in making the stability of an equilibrium point and the regulation to zero of an output robust to small unmodelled discrepancies between design model and actual system in particular those introducing an offset we propose a novel procedure which is intended to be relevant to realistic scenarios as illustrated by a non academic example

inverse spectral problem for quantum graphs the inverse spectral problem for the laplace operator on a finite metric graph is investigated it is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2 to prove the result a trace formula connecting the spectrum of the laplace operator with the set of periodic orbits for the metric graph is established

observer design for mimo non uniformly observable systems the design of high gain observers is usually based on normal forms of observability if the system is observable for every input uniform observability the gain of the observer does not required a solution to differential equation for multiple input multiple output mimo non uniformly observable systems we give here a sufficient condition that the input must satisfy in order to design an observer unlike uniformly observable systems the observer gain of non uniformly observable systems is derived from a lyapunov differential equation

fuzzy proximities and totally bounded fuzzy uniformities after the classical papers of zadeh 181 and chang 3 many concepts of general topology have been extended to fuzzy set theory in this paper we are interested to develop the study of fuzzy uniformities fuzzy proximities and of the connexions between such structures fuzzy uniformities have been introduced by lowen in 9 and with slight modifications in lo and by hutton in 5 the two approaches are quite different however the one proposed by hutton seems to suit much better to fuzzy set theory also because it allows to generalize a lot of results in the most proper way therefore in this paper we shall deal with hutton fuzzy uniformities the concept of fuzzy proximity used up to now see e g 7 8 15 1 is quite unsatisfactory indeed its fuzzyness is rather poor since these fuzzy proximities are in a canonical l 1 correspondence with the usual proximities see remark 2 6 moreover the open sets of the induced topologies are crisp and though every lowen fuzzy uniformity induces a fuzzy proximity this correspondence cannot work well since the two structures do not give the same fuzzy topology for these reasons we propose a new definition of fuzzy proximity it differs from the old one only in axiom p5 in short we could say that we privilege complementation with respect to intersection an agreement between the two operations being problematic in fuzzy set theory we point out that as a consequence a fuzzy set may be far from itself however this fact ought not to amaze anyone since it may happen just for pale sets which in a certain sense resemble the empty set nevertheless this modification enables us to associate a topology in a completely different way moreover every fuzzy uniformity induces a fuzzy proximity and vice versa we succeed to construct a fuzzy uniformity starting with a given fuzzy proximity and the induced topologies do not change at any step this implies that the topologies which admit a fuzzy proximity are exactly the completely regular ones moreover we prove that there exists a l l correspondence between fuzzy proximity spaces and a

fuzzy boundary and characteristic properties of order homomorphisms abstract in this paper first the concept of the boundary of an element in a fuzzy topological space is defined and studied then the concepts of boundary preserving interior preserving and closure preserving order homomorphisms etc are introduced and discussed finally the relations among these are obtained

the number of nodal domains on quantum graphs as a stability index of graph partitions the courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of laplacian type operators in particular it holds for generic eigenfunctions of a quantum graph the theorem stipulates that after ordering the eigenvalues as a non decreasing sequence the number of nodal domains n of the n th eigenfunction satisfies n n here we provide a new interpretation for the courant nodal deficiency d n n n in the case of quantum graphs it equals the morse index at a critical point of an energy functional on a suitably defined space of graph partitions thus the nodal deficiency assumes a previously unknown and profound meaning it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction to demonstrate this connection the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail

h λ completely hausdorff axiom on l topological spaces this paper defines the new concept of completely hausdorff axiom of an l topological space by means of l continuous mappings from an l topological space to the refined hutton s unit l interval by wang some characterizations of the completely hausdorff axiom defined in this paper are given and many nice properties of this kind of completely hausdorff axiom are proved for example it is hereditary and product invariant the refined hutton s unit l interval satisfy this kind of completely hausdorff axiom and when an l topological space satisfy this kind of completely hausdorff axiom every f convergent ideal does not have f limit points with different supports etc the relation between the completely hausdorff axiom defined in the paper and other separation axioms is discussed also

high gain observer for structured multi output nonlinear systems in this note we present two system structures that characterize classes of multi input multi output uniformly observable systems the first structure is decomposable into a linear and a nonlinear part while the second takes a more general form it is shown that the second system structure being more general contains several system structures that are available in the literature two high gain observer design methodologies are presented for both structures and their distinct features are highlighted

pointwise uniformities in fuzzy set theory a theory of pointwise quasi uniformities is based on fuzzy sets many theorems corresponding to the usual theorems in general topology are generalized on fuzzy sets in particular each fuzzy topological space is pointwise quasi uniformizable a pointwise uniformity is characterized in terms of a kind of complete regularity a uniformity on the fuzzy unit interval is constructed

pointwise topology on completely distributive lattices abstract in this paper the author presents a kind of pointwise topology theory on complete completely distributive lattices which is both a generalization of fuzzy topology and a further generalization of the author s papers 14 15

another look at order homomorphisms on fuzzes abstract order homomorphisms between fuzzy spaces are considered a general result is established which leads to an extension and simplification of earlier work by g j wang fuzzy sets and systems 12 1984 281 288

srn continuous order homomorphism and its characterizations in this paper the concept of semi regular continuity of order homomorphism is introduced and characterized by molecular nets and ideals in l topological spaces furthermore several properties of this continuity including hereditability and multiplicable properties are studied

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

smooth preuniform and preproximity spaces abstract fuzzy sets and fuzzy logic are very suited to interpret every day life sentences fuzzy interpretations are defined with the use of membership functions whose role is simply to give a kind of graphical visual support to help us to capture the meaning we think that generally membership values are not very important but only some characteristics of membership functions have to be kept we think that monotonic relations between degrees with which some properties are satisfied is an important characteristic we translate in this sense axioms corresponding to preuniform structures and preproximity structures contrary to fuzzy extensions like fuzzy topology for instance which generally keep the original axioms but adapt few of them to deal with fuzzy sets our approach induce much more transformations finally we study how classic constructions relating these concepts can be translated in this framework

separation axioms in l fuzzy topological spaces i t0 and t1 abstract using the notion of remote neighborhood we define the separation axioms t 0 and t 1 in l fuzzy topological spaces l fts the relations between our definitions hutton and reilly s and wang s are discussed and the separations of hutton s fuzzy unit interval and gantner s fuzzy real line are examined characterizations of these l fts are proved and a series of properties of them are investigated moreover some results on minimal t 0 l fts and t 1 l fts are established which will be used immediately in our subsequent discussion

sr convergence theory in fuzzy lattices abstract convergence theory and order homomorphisms theory are two important objects of study in fuzzy lattices in this paper sr convergence theory for molecular nets and ideals is established and the concepts of r irresolute sr continuous and almost irresolute order homomorphisms in topological molecular lattices are introduced and systematically characterized some mutual relations between these concepts and others are obtained

l fuzzy smirnov compactifications in this paper we get the pointwise characterization of the l fuzzy topology induced by an l fuzzy proximity space and study some important properties of l fuzzy proximity induced spaces also we discuss the n compactness separated relations and separation axioms of l fuzzy proximity spaces on the base of the method of induced spaces and above conclusions we establish the l fuzzy smirnov compactification and construct an example for l fuzzy smirnov compactifications

a high gain nonlinear observer with limited gain power in this note we deal with a new observer for nonlinear systems of dimension n in canonical observability form we follow the standard high gain paradigm but instead of having an observer of dimension n with a gain that grows up to power n we design an observer of dimension 2n 2 with a gain that grows up only to power 2

nonlinear observers with linearizable error dynamics we present a new method for designing asymptotic observers for a class of nonlinear systems the error between the state of the system and the state of the observer in appropriate coordinates evolves linearly and can be made to decay arbitrarily exponentially fast

on the linearization up to multi output injection for a class of systems with implicitly defined outputs for a class of nonlinear systems we investigate the problem of under what conditions there exists a coordinate transformation that yields a state affine linear system up to output injection with implicit outputs in particular we provide necessary and sufficient conditions for time varying linearization up to multi output injection we highlight that if the conditions hold as a consequence it is possible in the new coordinates to construct an observer with linear error dynamics we propose a methodology to find the coordinate transformation several examples illustrate the proposed procedure

fuzzy pretopological spaces and their representation today it is obvious that topology and uniformity and the many related theories have and will have a fundamental play in applied sciences every representation of real entities in a mathematical language necessarily implies a topological study of its goodness it is a problem of linguistic translation continuity as defined by hammer 71 this translation must preserved some important features of the original the power of actual digital computers enables us to use new ways in automatic pattern recognition but at the same time it shows us the many difficulties we encounter in searching a proper formulation of the topological concepts that everyone has in mind and uses in a naturally and more or less conscientious fashion one of the first axiomatisations of the concept of topological space was done by hausdorff and it is not so old it is only recently that some authors have thought if necessary to generalize these concepts and have obtained practical applications for instance appert 11 hammer 6 71 brissaud 3 mougeot auray duru 2 this has been especially motivated by the necessity of finding new concepts which well apply to finite sets and approx imations as hammer remarks topologists actually had one model in ind the real continuum and its generated euclidean spaces but not approximation and certainly not finite sets we think that the effort of interpretation and application to many fields such as linguistics system theory and education made by hammer is of great importance 7 and its references an important discovery in system theory which has been inforpally stated as a principle by zadeh 111 is as the complexity of a system increases our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics this principle is very close to limit problems and paradoxes encountered by physicians in particular the similitude with the heisenberg uncertainty prin ciple is clear it is certainly a motivation for the use of fuzzy set theory but these two aspects pretopology or extended topology as named by hammer

on induced fuzzy topological spaces the concept of induced fuzzy topological spaces was introduced by weiss 11 subsequently lowen 2 31 defined two mappings o and i which established the relationship between fuzzy topological spaces and topological spaces and proved some deeper results on induced fuzzy topological spaces since then several other authors continued the investigation of such spaces in this paper we will use the decomposition theorem for fuzzy sets to simplify the proofs of some theorems due to lowen and discuss the closure and interior operators separability ci c and separation properties of induced fuzzy topological spaces

extended high gain observer design for a class of mimo non uniformly observable systems abstract this paper deals with the design of an extended high gain observer ehgo for a large class of multiple input multiple output mimo non uniformly observable systems indeed the involved system nonlinearities does not assume the usual triangular structure as it is often the case when a high gain approach is adopted this structure constraint relaxation is achieved by an appropriate use of the characteristic indices associated to the state components there are two fundamental features that are worth to be pointed out firstly the gain of the proposed ehgo is provided by an appropriate riccati ordinary differential equation ode secondly the convergence of the observation error is guaranteed under some persistent excitation condition which can be checked on line simulation results are given to illustrate the performance of the proposed ehgo

a characteristic equation associated to an eigenvalue problem on c2 networks abstract a characteristic equation for the spectrum of an eigenvalue problem on c 2 networks is deduced and those special cases are investigated in which the spectrum depends mainly on the structure of the abstract graph belonging to the c 2 network

fuzzy sets and de morgan algebras abstract this paper examines some relations between on the one hand the concept of fuzzy sets and fuzzy topologies and on the other hand the concept of complete de morgan algebras and the topologies associated herewith

observer design for nonlinear triangular systems with unobservable linearisation the article deals with the observer design problem for a wide class of triangular time varying nonlinear systems with unobservable linearisation sufficient conditions are derived for the existence of a luenberger type observer when it is a priori known that the initial state of the system belongs to a given nonempty bounded subset of the state space for the general case the state estimation is exhibited by means of a switching sequence of time varying dynamics

minimal hausdorff l fuzzy topological spaces abstract in this paper the author introduced the concept of minimal hausdorff l fuzzy topological spaces investigated basic properties of it by means of fuzzy ideals and fuzzy nets and compared it with the concepts of compactness and h closedness

theory of topological molecular lattices abstract the aim of this paper is to bring the theories of point set topology fuzzy topology and l fuzzy topology into a unified theory the theory of topological molecular lattices for this purpose we first introduce the concepts of molecule for completely distributive complete lattice molecular lattice for short the cotopology on it and the concept of remote neighborhoods of molecules therefrom we develop the theory of pointwise topology on molecular lattices

a sufficient and necessary condition for fuzzy functions to be zadeh type functions abstract in this paper for an arbitrary fuzz l a sufficient and necessary condition for fuzz functions to be zadeh type functions is given

some results on i l and r l abstract this paper gives a partial solution to problem 1 in wang and lu 1992 and a suggestion about how to construct a better fuzzy unit interval it defines a new kind of fuzzy real line and investigates a series of its properties

generalization of inductive topologies to l topological vector spaces in this paper using the theory of l topological vector spaces we study some properties of inductive topologies determined by the family of l fuzzy linear order homomorphisms defined by fang jin xuan j fuzzy math 4 1 1996 93 and give a characterization of inductive topologies determined by a single l fuzzy linear order homomorphism as an example we point out the quotient l topology of l tvs is an inductive topology

local observer for infinitesimally observable nonlinear systems in this article we present a local observer based on infinitesimal observability along trajectories the gain of this observer derives from a dynamical lyapunov equation the inputs for which the observer converges are those which render the system observable at each point of the limit of the unknown trajectory infinitesimal observability at infinity along trajectories the performance of the proposed observer is illustrated with an application to dc machine

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

smooth topological spaces abstract in 1986 badard introduced the concept of a smooth topological space in this paper we give some links between smooth topological spaces and the corresponding results in chang s fuzzy topological spaces the concepts of subspace continuity and compactness are studied

smooth pretopogenous structures abstract in this paper we present smooth extension of pretopogenous structure which is obtained by weakening the axioms of the corresponding classical structure this weakening consists in the embedding of the structure in a multiple valued logic context a natural link is established between these concepts and the concept of fuzzy pretopological spaces smooth preproximity spaces and fuzzy preuniform spaces the notation of product smooth pretopogenous structures is also introduced

a luenberger like observer for nonlinear systems a state observer is proposed for nonlinear continuous time systems which extends the well known luenberger observer in particular on the basis of simple assumptions on the regularity of the system equations observability and the global holder condition for suitable functions which are generally satisfied for physically meaningful dynamic systems the global asymptotic convergence of the estimated state towards the true state is shown finally some examples of applications are also reported showing the effectiveness of the proposed observer

schrodinger operators on graphs and geometry i essentially bounded potentials the inverse spectral problem for schrodinger operators on finite compact metric graphs is investigated the relations between the spectral asymptotics and geometric properties of the underlying graph are studied it is proven that the euler characteristic of the graph can be calculated from the spectrum of the schrodinger operator in the case of essentially bounded real potentials and standard boundary conditions at the vertices several generalizations of the presented results are discussed

h l completely regular l fuzzy sets and their applications first we introduce the concept of h completely regular l fuzzy set in l fts and give some characterizations of it then we propose the concept of h complete regularization of l fts and investigate some properties of this complete regularization we also establish the relations between h completely regular l fts completely regular l fts and weakly induced l fts based on these results we obtain several characterizations of h completely regular l fts

fuzzy preuniform structures and the structures they induce 1 main results with the increasing complexity of the systems that men have created or have tried to understand it has appeared that the classic quantitative tools are not well adapted we must take into account the imprecision inherent to every modelling of complex systems on this problem zadeh 131 stated the following principle as the complexity of a system increases our ability to make precise and yet significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics for this reason qualitative tools have been developed topological and connected theories qualitative dynamics etc and new theories have been built up fuzzy sets and possibility theories in fact the modelling problem is a continuity like one in the sense that it must carry and preserve certain important features of the original real world it is certainly this problem that thorn has in mind when he says that qualitative dynamics consiste a admettre a priori l existence d un modele differentiel sous jacent au processus etudie et faute de connaitre explicitement ce modele a deduire de la seule supposition de son existence des conclusions relatives a la nature des singularitis du processus 121 in this paper we generalize the concept of uniform space by defining preuniform spaces and a typology on them at the same time we bring this theory together with that of fuzzy sets in order to get a very powerful tool we have used the zadeh definition of fuzzy sets but the major part of our results can be translated into the case of l fuzzy sets when

on smooth preproximity spaces abstract in this paper we introduce some properties of fuzzy smooth preproximity spaces and we study the cuts of fuzzy smooth preproximity spaces the product and sum of these spaces are given

q convergence of ideals in fuzzy lattices and its applications abstract in this paper we study q convergence and q convergence of ideals in fuzzy lattices by the concept of q remote neighborhood then we introduce the strong s irresoluteness s irresoluteness and s strong semicontinuity on fuzzy lattices we also study some properties of the notions above and strong semicontinuity 2 3 and s irresoluteness 3 4 with the aid of the q convergence q convergence of ideals

the characterizations of semi continuous and irresolute order homomorphisms on fuzzes abstract the concepts of semi continuous and irresolute order homomorphisms on fuzzes were introduced in j jianghan petroleum institute 2 1988 96 97 in this paper we first present the theory with respect to s convergence s convergence of molecular nets and ideals on fuzzes secondly we introduce the notions of strong semicontinuity strong irresoluteness and s irresoluteness on fuzzes then we systematically study the characterizations of these notions with the aid of the s convergence s convergence of molecular nets and ideals and give the relations of them

observability and observers for nonlinear systems this paper deals with observability and observers for general nonlinear systems in the non control affine case we characterize systems that are observable independently of the inputs an exponential observer for these systems is also exhibited

order homomorphisms on fuzzes abstract in this paper basic properties of the concept of order homomorphisms on fuzzes 4 which is a proper generalization of the concept of fuzz functions 1 are studied conditions for fuzz functions to be functions of zadeh s type are obtained moreover the concepts of continuous closed and open order homomorphisms are introduced and studied

nodal statistics on quantum graphs it has been suggested that the distribution of the suitably normalized number of zeros of laplacian eigenfunctions contains information about the geometry of the underlying domain we study this distribution more precisely the distribution of the nodal surplus for laplacian eigenfunctions of a metric graph the existence of the distribution is established along with its symmetry one consequence of the symmetry is that the graph s first betti number can be recovered as twice the average nodal surplus of its eigenfunctions furthermore for graphs with disjoint cycles it is proven that the distribution has a universal form it is binomial over the allowed range of values of the surplus to prove the latter result we introduce the notion of a local nodal surplus and study its symmetry and dependence properties establishing that the local nodal surpluses of disjoint cycles behave like independent bernoulli variables

concrete method for recovering the euler characteristic of quantum graphs trace formulas play a central role in the study of spectral geometry and in particular of quantum graphs the basis of our work is the result by kurasov which links the euler characteristic of metric graphs to the spectrum of their standard laplacian these ideas were shown to be applicable even in an experimental context where only a finite number of eigenvalues from a physical realization of quantum graph can be measured in the present work we analyse sufficient hypotheses which guarantee the successful recovery of we also study how to improve the efficiency of the method and in particular how to minimise the number of eigenvalues required finally we compare our findings with numerical examples surprisingly just a few dozens of eigenvalues can be enough

observer design for uncertain nonlinear systems with unmodeled dynamics this paper is concerned with the observer design problem for a class of multi input multi output nonlinear systems with the unmodeled dynamics unknown parameters and external disturbance a dynamic signal which can dominate the unmodeled dynamics is firstly constructed then two types of observer schemes that is adaptive observer and robust observer are respectively proposed the observation error in the two schemes can be made arbitrarily small by choosing the appropriate design parameters an illustrative example is provided to demonstrate the validity of the proposed design methods

fuzzy proximity spaces redefined abstract we use the concept l quasi coincident in fuzzy set theory to redefine l fuzzy proximity space and we study the following problems proximity of l fuzzy normal spaces l fuzzy filters proximity neighborhoods and the l fuzzy topology induced by the l fuzzy proximity

uniformity and proximity of l fuzzy topological spaces abstract in 3 4 a k katsaras defined fuzzy proximity spaces and studied some of their properties in 5 we used the concept l quasi coincident in fuzzy set theory and fuzzy topology to redefine l fuzzy proximity space in such a way that it takes fuzzy proximity space as a special case also some pathologic properties were avoided for example proposition 2 1 in 4 in 5 we proved also some results analogous to those that hold for ordinary proximity spaces in this paper we continue the study of l fuzzy proximity spaces we shall prove that the partially ordered set x of all l fuzzy proximities on x is a complete lattice theorem 2 5 and we shall discuss the relationship between l fuzzy uniform spaces and l fuzzy proximity spaces theorem 3 4 4 1