quasi coincidence and quasi fuzzy hausdorff the concept of quasi coincidence of a fuzzy point to a fuzzy set is investigated the notion of a quasi fuzzy hausdorff topological space is introduced and its close relationship with the properties of fuzzy hausdorff and hausdorf is shown

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

a classification of types of extensions from top to qftop abstract by making use of the concept of the introduced quasi fuzzy topological space c y zheng fuzzy sets and systems 14 1984 273 280 we introduce the concept of s extensions mainly we generalize and characterize the concepts of good extensions r lowen j math anal appl 56 1976 621 633 weak inducement h w martin j math anal appl 78 1980 634 639 and s extensions by introducing some relations between the category of topological spaces and the category of quasi topological spaces

paracompactness in l topological spaces in this paper a certain fuzzy paracompactness is presented by means of the concept of locally finite family of l fuzzy sets the concept of locally finite family can be used to characterize fuzzy countable compactness this fuzzy paracompactness is a natural generalization of lowen fuzzy compactness a fuzzy countably compact l set is fuzzy paracompact if and only if it is fuzzy compact a topological space x j is paracompact if and only if its induced space x j is fuzzy paracompact when ms l the fuzzy enhanced real line r l is fuzzy paracompact the product of a fuzzy compact set and a fuzzy paracompact set is fuzzy paracompact a fuzzy strongly t2 paracompact space is strongly s regular and strongly ws normal moreover a strongly s regular lindelof space is fuzzy paracompact

on consistency and ranking of alternatives in fuzzy ahp abstract in fuzzy ahp where pairwise comparisons are described by membership values the treatment of consistency has largely been sketchy this paper proposes a fuzzy consistency definition with consideration of a tolerance deviation essentially the fuzzy ratios of relative importance allowing certain tolerance deviation are formulated as constraints on the membership values of the local priorities the fuzzy local and global weights are determined via the extension principle the alternatives are ranked on the basis of the global weights by application of a maximum minimum set ranking method our analysis uses only the data which are considered as consistent the proposed approach is computationally efficient as it consists of solving only linear programs an example is included for illustrative purpose

data driven fuzzy preference analysis from an optimization perspective abstract how to leverage massive online data to understand consumer preferences over products and services has accumulated significant attention in business in this paper we analyze consumer preferences by modeling it as a decision making problem of ranking alternatives with consumers online ratings we propose a data driven fuzzy preference analysis d fpa method to obtain the priorities of alternatives we show that the d fpa is tractable and with high computation efficiency in addition we propose a natural indicator to measure the reliability of the derived ranking results and suggest thresholds of this indicator for better control of the method a real world application about online film rating is provided to illustrate the d fpa demonstrating that the derived ranking results converge rapidly and remain stable with the observed empirical data finally we show how to build up an effective recommendation system with empirical data from movielens

fuzzy geometry an updated overview abstract this paper briefly reviews work on the fuzzy topology and geometry of image subsets including adjacency separation and connectedness distance and relative position area perimeter and diameter convexity medial axes and thinning as well as some applications of these concepts in image processing and analysis

fuzzy completely irresolute and fuzzy weakly completely irresolute functions abstract the aim of this paper is to introduce two new classes of functions called fuzzy completely irresolute and fuzzy weakly completely irresolute functions their characterizations examples compositions of these functions their relationships with other functions and preservations of some fuzzy spaces under these functions are studied

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

fuzzy pairwise extremally disconnected spaces abstract in this paper we generalize the concept of fuzzy extremally disconnected spaces due to ghosh 1992 into a fuzzy bitopological setting and investigate some of its properties using the concepts of i j semi closure i j closure and related notions in a fuzzy setting

fuzzy initial convergence structure abstract we introduce the notion of fuzzy initial convergence structure induced by family of functions the notion of fuzzy initial convergence structure is shown to be a generalization of fuzzy product convergence structure

the spectral gap for some spin chains with discrete symmetry breaking we prove that for any finite set of generalized valence bond solid gvbs states of a quantum spin chain there exists a translation invariant finite range hamiltonian for which this set is the set of ground states this result implies that there are gvbs models with arbitrary broken discrete symmetries that are described as combinations of lattice translations lattice reflections and local unitary or anti unitary transformations we also show that all gvbs models that satisfy some natural conditions have a spectral gap the existence of a spectral gap is obtained by applying a simple and quite general strategy for proving lower bounds on the spectral gap of the generator of a classical or quantum spin dynamics this general scheme is interesting in its own right and threfore although the basic idea is not new we present it in a system independent setting the results are illustrated with a number of examples

examining alternatives in the interval analytic hierarchy process using complete enumeration the analytic hierarchy process as an aid to decision makers has gained in popularity over the years recently the procedure has been generalised into the interval analytic hierarchy process rather than select a precise ratio for the comparisons between the alternatives offered to the decision maker an interval is proposed one popular strategy in examining this problem is to adopt a simulation technique here two simulation methods are considered and it is demonstrated that total enumeration is a viable alternative

connectedness of coherent fuzzy pictures abstract a fuzzy digital picture is called connected if it has only one connected component of constant fuzzy level that is maximal i e not adjacent to any component of higher fuzzy level in this paper we propose two acceptors defined over fuzzy digital pictures one is a one pebble acceptor and another is a deterministic one way nonwriting parallel sequential acceptor we also prove that both acceptors can determine whether a given coherent fuzzy digital picture is connected

analytic hierarchy process hesitant group decision making in this paper we consider that the judgments provided by the decision makers dms cannot be aggregated and revised then define them as hesitant judgments to describe the hesitancy experienced by the dms in decision making if there exist hesitant judgments in analytic hierarchy process group decision making ahp gdm then we call it ahp hesitant group decision making ahp hgdm as an extension of ahp gdm based on hesitant multiplicative preference relations hmprs to collect the hesitant judgments we develop a hesitant multiplicative programming method hmpm as a new prioritization method to derive ratio scale priorities from hmprs the hmpm is discussed in detail with examples to show its advantages and characteristics the practicality and effectiveness of our methods are illustrated by an example of the water conservancy in china

hesitant analytic hierarchy process abstract in traditional analytic hierarch process ahp decision makers dms are required to provide crisp judgments over paired comparisons of objectives to construct comparison matrices to enhance the modeling ability of traditional ahp we propose hesitant ahp h ahp that can consider the hesitancy experienced by the dms in decision h ahp is characterized by hesitant judgments where each hesitant judgment can be represented by several possible values different probability distributions can be used to further describe hesitant judgments according to the dms preferences based on a hesitant comparison matrix hcm that consists of hesitant judgments we define two indices to measure the consistency degree and the consensus degree of the hcm respectively from a stochastic point of view a new prioritization method is developed to derive priorities from hcms where the results are with probability interpretations we provide a step by step procedure for h ahp and demonstrate this new method with a real life decision making problem

a note on the articles strongly compact fuzzy topological spaces and fuzzy preopen sets and fuzzy preseparation axioms abstract in this note we show by producing counterexamples that some results which appeared in the articles by nanda 1991 and singal and prakash 1991 are incorrect

rs compactness in l fuzzy topological spaces abstract a good definition of rs compactness in l fuzzy topological spaces is suggested for l a fuzzy lattice we define rs compactness on arbitrary l fuzzy sets and study some of its properties

principal superconnected fuzzy topological spaces abstract in this note principal superconnected fuzzy topological spaces are introduced and it is proved that such a space is path connected

a sensitivity analysis algorithm for hierarchical decision models in this paper a comprehensive algorithm is developed to analyze the sensitivity of hierarchical decision models hdm including the analytic hierarchy process and its variants to single and multiple changes in the local contribution matrices at any level of the decision hierarchy the algorithm is applicable to all hdm that use an additive function to derive the overall contribution vector it is independent of pairwise comparison scales judgment quantification techniques and group opinion combining methods the allowable range region of perturbations contribution tolerance operating point sensitivity coefficient total sensitivity coefficient and the most critical decision element at a certain level are identified in the hdm sa algorithm an example is given to demonstrate the application of the algorithm and show that hdm sa can reveal information more significant and useful than simply knowing the rank order of the decision alternatives 2007 elsevier b v all rights reserved

fn closed sets and fuzzy locally nearly compact spaces abstract we define a fuzzy subset fn closed relative to i and study the special properties of such sets in fuzzy nearly compact spaces and in fuzzy hausdorff spaces the definition and characterizations of fuzzy locally nearly compact spaces are introduced finally effects on semi regularization of are discussed

on various weak fuzzy pairwise retracts the notion of a fuzzy retract was introduced by rodabaugh j math anal appl 79 1981 273 and this notion was extended to the fuzzy bitopological setting j fuzzy math 9 1 2001 207 in this paper we define the notions of fuzzy pairwise semiretracts fuzzy pairwise preretracts fuzzy pairwise strongly semiretracts fuzzy pairwise less strongly semiretracts and fuzzy pairwise semi preretracts we establish some properties of these concepts moreover the relations between these new concepts are discussed

connectedness on fuzzy topological spaces in this paper properties of connectedness on completely distributive topological lattices definition 2 1 and fuzzy topological spaces 1 are studied some equivalent propositions on the concepts of functions in completely distributive lattices and connected elements in completely distributive lattices and fuzzy topological spaces are proved specially connectedness is characterized both by the r neighborhoods of molecules and c separations for any family li i i i of fuzzy topological spaces the product 1 is connected iff every li i is connected a continuous function maps connected elements onto connected ones the connectedness on the fuzzy real line r l is discussed in the last section as an important example the intervals defined here definition 4 1 must be connected in the fuzzy real line r l and so the intervals defined by rodabaugh in 13 are too any connected element is connected in the fuzzy real line r l if and only if r l r

the diameter of a fuzzy set abstract in pattern recognition one often wants to measure geometric properties of imprecisely defined subsets of an image this paper proposes definitions of intrinsic and extrinsic diameter for fuzzy subsets which reduce to the ordinary definitions when the subsets are crisp we also define height and width for a fuzzy subset and show how they relate to the area i e integral of membership for convex fuzzy subsets the intrinsic diameter cannot exceed the extrinsic diameter but it can be smaller finally for piecewise constant convex fuzzy subsets the intrinsic diameter cannot exceed half the fuzzy perimeter but this need not be true in the nonconvex case

the analytic hierarchy process in an uncertain environment a simulation approach abstract traditionally decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the analytic hierarchy process ahp since many circumstances exist which make such a convergence difficult confidence in the outcome of an ensuing ahp synthesis may be reduced likewise when a group of decision makers cannot arrive at a consensus regarding a judgment some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments the ahp utilizes point estimates in order to derive the relative weights of criteria sub criteria and alternatives which govern a decision problem however when point estimates are difficult to determine distributions describing feasible judgments may be more appropriate using simulation we will demonstrate that levels of confidence can be developed expected weights can be calculated and expected ranks can be determined it will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis

on the topological properties of fuzzy rough sets this paper is devoted to the discussion of the relationship between fuzzy rough set models and fuzzy topologies on a finite universe the tc axiom for fuzzy topology is proposed it is proved that the set of all lower approximation sets based on a reflexive and transitive fuzzy relation consists of a fuzzy topology which satisfies tc axiom and conversely a fuzzy topology which obey tc axiom is just the set of all lower approximation sets under a reflexive and transitive fuzzy relation that is to say there exists a one to one correspondence between the set of all reflexive and transitive fuzzy relations and the set of all fuzzy topologies which satisfy tc axiom

theoretical estimation of the probability of weight rank reversal in pairwise comparisons abstract pairwise comparison is a key component in multi criteria decision making the probability of rank reversal is a useful measure for evaluating the impact of uncertainty on the final outcome in the context of this paper the type of uncertainty considered is related to the fact that experts have different opinions or that they may perform inconsistent pairwise comparisons we provide a theoretical model for estimating the probability of the consequent rank reversal using the multivariate normal cumulative distribution function the model is applied to two alternative weight extraction methods frequently used in the literature the geometric mean and the eigenvalue method we introduce a reasonable framework for incorporating uncertainty in the decision making process and calculate the mean value and cross correlation of the average weights which are required in the application of the model the theoretical results are compared against numerical simulations and a very good agreement is observed we further show how our model can be extended in applications of a full multi criteria decision making analysis such as the analytic hierarchy process we also discuss how the theoretical model can be used in practice where the statistical properties of the uncertainty induced perturbations are unknown and the only information provided by the pairwise comparison matrices of a small group of experts the methodology presented here can be used to extend the pairwise comparison framework in order to provide some information on the credibility of its outcome

some other types of fuzzy connectedness abstract in this paper we investigate level connectedness concepts in fuzzy topological spaces fts in short we introduce connectedness concepts e g c3 and c4 in fuzzy topology and establish their appropriateness through several observations

on fuzzy t continuous mappings abstract the concepts of fuzzy almost continuous fuzzy continuous and fuzzy weakly continuous mappings have been introduced by yalvac and by mukherjee and sinha the aim of this paper is mainly to study and to find the mutual interrelations among these concepts

a medial axis transformation for grayscale pictures blum s medial axis transformation mat for binary pictures yields medial axis points that lie midway between opposite borders of a region or along angle bisectors this note discusses a generalization of the mat in which a score is computed for each point p of a grayscale picture based on the gradient magnitudes at pairs of points that have p as their midpoint these scores are high at points that lie midway between pairs of antiparallel edges or along angle bisectors so that they define a mat like skeleton which we may call the gradmat however this skeleton is rather sensitive to the presence of noise edges or to irregularities in the region edges and it also is subject to artifacts created by pairs of edges belonging to different objects

fuzzy weak totally semicontinuous and fuzzy weak totally precontinuous mappings abstract the aim of this paper is to introduce some new classes of mappings called fuzzy weak totally continuous fuzzy totally precontinuous fuzzy weak totally semicontinuous fuzzy totally m precontinuous fuzzy totally m precontinuous and fuzzy totally irresolute mappings their compositions their relationships with other mappings preservations of some fuzzy spaces under these mappings and some examples are studied we also prove that the property that the mapping is fuzzy totally semicontinuous and fuzzy totally precontinuous is equivalent to the property that the mapping is fuzzy totally continuous

a min max medial axis transformation blum s medial axis transformation mat of the set s of 1 s in a binary picture can be defined by an iterative shrinking and reexpanding process which detects corners on the contours of constant distance from s and thereby yields a skeleton of s for unsegmented gray level pictures one can use an analogous definition in which local min and max operations play the roles of shrinking and expanding to compute a mmmat value at each point of the picture the set of points having high values defines a good skeleton for the set of high gray level points in the given picture

on some weaker forms of fuzzy continuous and fuzzy open mappings on fuzzy topology spaces abstract the purpose of the present paper is to study some weaker forms of fuzzy continuous and fuzzy open maps between fuzzy topological spaces along with the introduction and investigations of a new class of non fuzzy continuous maps this latter type of mappings is seen to be independent of the other non continuous functions discussed here all these functions are characterized and certain properties concerning them are proved finally a comparative study as to their mutual dependence and interrelations under different conditions is taken up

on a fuzzy theory based computer aided particle shape description abstract a central problem of particle shape analysis consists in the characterization of the shape of any natural of produced particle by a small number of parameters which are to be evaluated from a two dimensional picture of the particle e g from its projection we suggest using the grey tone picture obtained by a camera to compute a sympathy function with respect to a suitable family of planar shapes the computation is performed by means of the image processing equipment robotron a 6471 the sympathy function defines a fuzzy set on the parameter space the procedure is applied to a quartz particle

a note on fuzzy starshaped fuzzy sets fuzzy starshaped sets have starshaped levels sets and generalise fuzzy convex sets they have similar metric properties and lp metrics may be defined using gauge functions the induced metric spaces are shown to be locally compact for 1 p and compact sets are characterised

two new l fuzzy topologies on r l in this paper two new topologies 1 2 on r l are constructed using i we produced two l fuzzy topologies i on r l other than the l fuzzy topology t introduced by hutton we have proved that 1 is finer than 2 and 2 is finer than t 2 is the coarsest among the induced l fuzzy topologies which are finer than t we have shown that lr l i the induced space of r l i possesses many good properties such as stronger separation suitability etc we proved that closed interval a b l 2 a b l is closed connected and n compact the addition and multiplication defined on r l by rodabaugh are still jointly continuous

on some near fuzzy continuous functions between fuzzy topological spaces abstract the concepts of fuzzy almost strongly continuous fuzzy continuous fuzzy weekly continuous and fuzzy weakly continuous are introduced and studied in the light of the concept of q coincidence in a fuzzy setting together with the concept of q neighbourhoods the notions of fuzzy neighbourhoods and fuzzy neighbourhoods are defined to investigate the characterizations and properties of these functions finally a comparative study regarding the mutual interrelations among these maps along with fuzzy continuous maps is made

valence bond ground states in a frustrated two dimensional spin 1 2 heisenberg antiferromagnet we study a class of two dimensional spin 1 2 heisenberg antiferromagnets introduced by klein 1 in which the nearest neighbor term is supplemented by next nearest neighbor pair and four body interactions producing additional frustration for certain lattices including e g the hexagonal lattice we prove that any finite subset which admits a dimer covering has a ground state space spanned by valence bond states each of which consists only of nearest neighbor dimer singlet pairs we also establish linear independence of these valence bond states the possible relevance to resonating valence bond theories of high temperature superconductors is briefly discussed in particular our results apply both to regular subsets of the lattice and to subsets with static holes

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

on several types of degree of fuzzy compactness abstract sostak 1985 defined a fuzzy topology on a set x as a fuzzy subset of the family i x of fuzzy subsets of x satisfying some axioms the concept of the degree of compactness of a fuzzy set was first introduced by sostak 1985 in this paper we discuss the degrees of almost compactness near compactness countable compactness light compactness and strong compactness of a fuzzy set also we investigate the behavior of the degrees of compactness under several types of fuzzy continuous functions

semi continous and semi closed mappings and semi connectedness in fuzzy setting abstract the aim of the paper is mainly to study and to find different characterizations of semi continuous and semi closed mappings between fuzzy topological spaces the notion of fuzzy semi open sets is used to define fuzzy ti spaces i 0 1 2 and these spaces are investigated under fuzzy semi continuity finally fuzzy semi connectedness is introduced and studied to some extent

irresolute and lmost open functions between fuzzy topological spaces abstract the aim of this paper is to introduce and characterize the concept of fuzzy irresolute functions which is strictly stronger than the concept of fuzzy semi continuous functions and independent of the concept of fuzzy continuous functions the independence of the two definitions of fuzzy almost open functions as introduced in 5 and 3 and some of their characterizations are also established

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

image segmentation using fuzzy correlation a concept of correlation between two properties fuzzy representations of an image is introduced a set of algorithms for image segmentation both fuzzy and nonfuzzy has been formulated the spatial information is taken care of by the following measures transitional correlation and within class correlation a relation between the correlation coefficient and the index of fuzziness is theoretically established and experimentally verified the effectiveness of the algorithms is illustrated on images having different types of histograms

loop coverings of two dimensional square lattices abstract in a two dimensional square lattice where all sites are covered by loops in the past few years a classical statistical mechanics model of dense loop gas was considered for its relevance for the evaluation of observables of the quantum antiferromagnetic heisenberg model on the resonating valence bond state by use of transfer matrices we evaluate several observables for narrow strips with a width of two or three sites a simple extrapolation consistent with rigorous bounds is conjectured for its possible use in a variational ansatz in the hubbard model

on nearly compact and 0 rigid fuzzy sets in fuzzy topological spaces abstract this paper intends to initiate the study of nearly compact and rigid fuzzy sets in a fuzzy topological space these two classes of fuzzy sets are seen to be properly contained in the class of fuzzy compact sets and properly contain the class of fuzzy almost compact sets several characterizations and certain other properties of these two types of fuzzy sets are established by using the concepts of fuzzy closures and closures of fuzzy sets and fuzzy filterbases

fuzzy sets and functions on fuzzy spaces abstract we generally follow the terminology of azad j math anal appl 82 1981 14 32 and ming j math anal appl 76 1980 571 599 in addition to the fundamental concepts for fuzzy sets we emphasize the usefulness of the concepts of fuzzy point fuzzy elementhood for fuzzy sets a and b a new characterization is given for relations a b and a b this knowledge permits us to combine the two definitions of fuzzy point fuzzy elementhood in the third section some results are given concerning various special types of fuzzy sets in fuzzy topological spaces and the fuzzy semi regular fuzzy regular spaces defined by azad in the last section the definitions of h almost continuous w almost open functions urysohn space which are defined by hussain wilansky and noiri respectively are extended to fuzzy sets furthermore some results are obtained in the functions of the fuzzy topological spaces defined by azad and those are defined here

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

fuzzy weak semicontinuity abstract in this paper we introduce the fuzzy weakly semicontinuous mapping which is the weaker form of both fuzzy semicontinuous and fuzzy almost continuous mappings in fuzzy topological spaces however fuzzy weakly semicontinuous mapping and fuzzy weakly continuous mapping are independent notions then we discuss some characteristic properties of fuzzy semicontinuous fuzzy semiopen fuzzy semiclosed and fuzzy pre semi open mappings

fuzzy extremally disconnected spaces abstract an investigation of extremally disconnected fuzzy topological spaces is done in this paper such a class of spaces is characterized in various ways by using the concepts of q neighborhoods semi q neighbourhoods closure closure semi closure and related notions in fuzzy setting

connectedness in fuzzy topological spaces abstract we introduce four notions of connectedness for an arbitrary fuzzy set and discuss all the implications that exist between them using these notions various equivalence relations are defined on the set of fuzzy points components and quasi components are formed as the union of certain equivalence classes a fuzzy space is shown to be a dispoint union of its components in all the four cases and non overlapping union of its quasi components

reciprocal distributions in the analytic hierarchy process abstract the paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the analytic hierarchy process ahp these distributions called reciprocal satisfy the reciprocity axiom of the ahp methodology first we define and characterise the reciprocal distributions and present some results and properties associated with them we then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector considering the eigenvector method egv in the consistent case and the row geometric mean method rgm as priorization methods and assuming that the judgements follow reciprocal distributions we prove that the elements of the priorities vector are distributed according to reciprocal random variables the knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems the probability that a given alternative would be ranked in the first position and the probability of a given ranking preference structure for the whole set of alternatives in particular we find the analytic expressions for these probabilities when a reciprocal random variable such as the lognormal is used

l topological meet theorems kindler gave a purely topological condition for an intersection of subsets of some set to be nonempty which plays an important role in topology this paper generalizes kindler s theorems to the fuzzy lattice based setting due to a proper choice of compactness and connectedness axioms it shows that almost all basic results of kindler survive in the extended setting

on fuzzy pairwise semi precontinuity abstract in this paper we generalize the concepts of fuzzy semi preopen semi preclosed sets and fuzzy semi precontinuous mappings due to park and lee fuzzy sets and systems 67 1994 359 364 into fuzzy bitopological setting and discuss some of its basic properties

fuzzy convergence structures abstract we define a fuzzy convergence structure which is a correspondence between the prefilters on a given set x and the fuzzy subsets of x the notion of fuzzy convergence structure is shown to be a generalization of a fuzzy topology

some good l fuzzy compactness related concepts and their properties i the aim of this paper is to introduce good definitions of almost light and near compactness in l fuzzy topological spaces where l is a fuzzy lattice these weak compactnesses are defined for arbitrary l fuzzy sets and their properties studied

on fuzzy weakly semi continuous functions abstract this paper is devoted to the introduction and study of fuzzy weakly semi continuous functions between fuzzy topological spaces some properties of these functions are characterized in terms of quasi coincidence quasi neighborhoods neighborhoods etc furthermore some relations connecting fuzzy weakly semi continuous functions and fuzzy retractions have been established

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

connectedness in l fuzzy topological spaces based on the consideration to the layer structures of fuzzy lattices and the level topologies of l fuzzy topological spaces a connectedness is defined for an arbitrary l fuzzy set in this paper this definition reflects the degree of connectivity of an l fuzzy set it is shown that the connectedness of l fuzzy topological spaces is an l good extension multiplicative and preserved under continuous l valued zadeh functions and that the inverse limit of continuums is a continuum general l fuzzy intervals and h intervals are defined and connectedness of them are proved it is also shown that the connectedness of an l fuzzy topological space is equivalent to the connectedness of its induced i l fuzzy topological space

fuzzy s continuous functions and semilocally connected fuzzy topological spaces abstract the concept of s continuous functions kohli 1978 is translated in fuzzy setting into two concepts fuzzy s continuous functions and fuzzy os continuous functions the concepts of semiconnected functions kohli 1978 and semilocally connected spaces kohli 1978 do the same interactions between those concepts in fuzzy setting and fuzzy continuity chang 1968 are discussed also characterizations of semilocally connected and semilocally o connected fuzzy topological spaces in terms of fuzzy s continuous functions and fuzzy os continuous functions respectively are obtained

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

some results on fuzzy digital convexity abstract the definition of fuzzy convexity is reviewed and some results on projections of convex and fuzzy convex sets are established digital fuzzy convexity is defined and relationships among alternative definitions are investigated

fuzzy path and fuzzy connectedness abstract this paper discusses fuzzy connectedness in fuzzy topology zadeh s classical paper 1 has introduced the fundamental concept of a fuzzy set upon which the foundation of fuzzy mathematics rests the theory of fuzzy topology is a branch of such mathematics our purpose is to introduce some basic concepts as fuzzy path fuzzy path connected set locally fuzzy path connected set in the fuzzy topological spaces and to prove some fundamental properties related to fuzzy connectedness

fuzzy neighborhood spaces abstract in this paper we introduce a subcategory of the category of fuzzy topological spaces and we prove basic properties of this subcategory the main idea behind these spaces is the notion of fuzzy neighborhoods which enables us to introduce a local theory of fuzzy topologies

a new thinning algorithm for gray scale images by the relaxation technique abstract a new thinning algorithm for gray scale images based on the relaxation technique is proposed thinning is treated as a process of assigning each picture point to either a skeleton or a nonskeleton category so that relaxation can be used to perform thinning contextual information existing in input data is utilized in the thinning process so that line straightness can be preserved five classes are created to classify each point with four classes belonging to the skeleton category for different orientations and one to the nonskeleton category an initial probability value for assigning a point to each class is computed first in the relaxation process interactions take place among the probability values of neighbouring points making the skeleton probability values of the points in compatible orientations to enhance one another the probability value for each point to belong to the nonskeleton class of each point is also refined according to the point type the relaxation process is terminated when it converges to a condition in which points lying on skeletons achieve high skeleton probability values while other points have unity nonskeleton probability values experimental results show that the proposed approach produces good thinning results in keeping line straightness

some stronger forms of fuzzy continuous mapping on fuzzy topological spaces abstract the paper is devoted to the introduction and study of the concepts of certain classes of functions between fuzzy topological spaces each of these functions presents a stronger form of the fuzzy continuous functions these maps have been characterized and investigated mainly in the light of the notions of q neighbourhoods quasi coincidence fuzzy closure and ds neighbourhoods they have also been compared as to their mutual dependence as also with regard to their interactions with certain other fuzzy non continuous functions

fuzzy strongly semiopen sets and fuzzy strong semicontinuity abstract in this paper we first introduce and study the fuzzy strongly semiopen and fuzzy strongly semiclosed sets in fuzzy space then we introduce the strong semi interior and strong semiclosure of a fuzzy set and kuratowski s 14 sets theorem of general topology is generalized finally we introduce the fuzzy strongly semicontinuous fuzzy strongly semiopen and fuzzy strongly semiclosed mappings on fuzzy topological spaces and also establish some of their characteristic properties and discuss relations between fuzzy continuous fuzzy semicontinuous and fuzzy strongly semicontinuous mappings

fuzzy completely pre irresolute and weakly completely pre irresolute mappings in this paper we introduce fuzzy completely pre irresolute and fuzzy weakly completely pre irresolute mappings between fuzzy topological spaces and study these mappings in relation to some other types of already known mappings

the analytic hierarchy process with stochastic judgements the analytic hierarchy process ahp is a widely used method for multicriteria decision support based on the hierarchical decomposition of objectives evaluation of preferences through pairwise comparisons and a subsequent aggregation into global evaluations the current paper integrates the ahp with stochastic multicriteria acceptability analysis smaa an inverse preference method to allow the pairwise comparisons to be uncertain a simulation experiment is used to assess how the consistency of judgements and the ability of the smaa ahp model to discern the best alternative deteriorates as uncertainty increases across a range of simulated problems results indicate that according to conventional benchmarks judgements are likely to remain consistent unless uncertainty is severe but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear

fuzzy faintly continuous functions abstract the aim of this paper is to introduce the concepts of fuzzy faintly continuous functions these functions have been characterized and investigated mainly in the light of the notions of quasi concidence q neighbourhoods fuzzy interior and fuzzy closure it is seen that fuzzy continuity implies fuzzy faintly continuity but not conversely the converse is also true if the co domain of the function is a fuzzy regular space finally a comparative study regarding the mutual interrelations among the fuzzy r map fuzzy completely continuous fuzzy almost continuous and fuzzy continuous functions along with fuzzy faintly continuous functions is made