geometric properties of domains related to μ synthesis abstract in the paper we study the geometric properties of a large family of domains called the generalized tetrablocks related to the synthesis containing both the family of the symmetrized polydiscs and the family of the 1 n quotients e n n 2 introduced recently by g bharali it is proved that the generalized tetrablock cannot be exhausted by domains biholomorphic to convex ones moreover it is shown that the caratheodory distance and the lempert function are not equal on a large subfamily of the generalized tetrablocks containing i a e n n 4 we also derive a number of geometric properties of the generalized tetrablocks as well as the 1 n quotients as a by product we get that the pentablock another domain related to the synthesis problem introduced recently by j agler z a lykova and n j young cannot be exhausted by domains biholomorphic to convex ones

a unifying study between modal like operators topologies and fuzzy sets the paper presents the essential connections between modal like operators topologies and fuzzy sets we show for example that each fuzzy set determines a preorder and an alexandrov topology and that similar correspondences hold also for the other direction further a category for preorder based fuzzy sets is defined and it is shown that its equivalent subcategory of representatives is isomorphic to the categories of preordered sets and alexandrov spaces moreover joins meets and complements for the objects in this category of representatives are determined this suggests how to define for fuzzy subsets of a certain universe the lattice operations in a canonical way

alternative characterizations for the representation of families of sets by fuzzy sets we investigate the representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets this concept of representation is widely used in the literature for the classification and study of fuzzy structures on the same underlying set negoita and ralescu obtained two theorems both giving necessary and sufficient conditions one called the representation theorem and the second called the generalized representation theorem our aim in this paper is to take a closer look at both theorems hence establishing some equivalent forms and several constructive remarks including some directions for future investigations

general form of lattice valued fuzzy sets under the cutworthy approach in this note a new solution of problem of synthesis of fuzzy sets is presented in other words necessary and sufficient conditions are formulated under which for a given family of subsets f of a set x and a fixed complete lattice l there is a fuzzy set m x l such that the collection of cuts of m coincides with f moreover it is proved that the general form of lattice valued fuzzy sets considering families of cuts are the type of fuzzy sets having the codomain 0 1 c for a suitable chosen cardinal c

on an application of fuzzy relations in biogeography abstract starting from a quasi ordering relation on a set by a well known construction an equivalence relation could be made as well as an ordering relation on the quotient set in this paper we give a similar fuzzy construction with lattice valued fuzzy relations if is a crisp correspondence from set a to set b then a quasi ordering relation on the set b could be defined in the following way x y if and only if z a z x z y in this paper an analogous fuzzy construction is given for a special type of fuzzy correspondences which appear in biogeography from this correspondence a quasi ordering relation is obtained as well as the corresponding fuzzy similarity relation and fuzzy ordering relation on the set of similarity classes this construction was developed to be applied in solving some problems of determining connections of different biogeographical areas according to recorded species here those results have been applied to determine connections of biogeographical territories in the balkan peninsula according to different species of genus cheilosia meigen 1822 and related genera diptera syrphidae hover flies

on p fuzzy correspondences and generalized associativity abstract a p fuzzy correspondence is a mapping a a 1 a 2 a n p where a1 a2 an are nonempty sets and p a partially ordered set in this paper we give necessary and sufficient conditions under which a family of ordinary correspondences on sets a1 a2 an is a family of levels of a p fuzzy correspondence a g groupoid is a quadruple s1 s2 s3 a where s1 s2 s3 are sets and a mapping a s1 s2 s3 we consider a type of p fuzzy correspondences as a generalization of a notion of g groupoids called pg fuzzy correspondences that is a pg fuzzy correspondence is a mapping a a 1 a 2 a 3 p where all sets a1 a2 an and p are partially ordered here we deal with binary pg fuzzy correspondences we give a general solution of the functional equation of generalized associativity a x b y z c d x y z where a b c and d are binary pg fuzzy correspondences in particular if a b c and d are semilattices which are solutions of the considered equation then it turns out that they are equal i e a b c d

existence and uniqueness of fuzzy ideals let r be a commutative ring the idea of identifying a fuzzy ideal of r with its corresponding chain of level ideals and its range of values although widely used in the literature for the study of fuzzy ideals has never been investigated in a general setting before we investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established in particular we obtain that uniqueness always holds true for artinian rings moreover examples are presented to further support our results and shed more light on the above mentioned problem of identification we finish by raising a natural question that is left as an open problem for further investigation

relational valued fuzzy sets abstract a generalization of the notion of a fuzzy set is given an r fuzzy set a is a mapping from a set a into the relational system s where is a binary relation on s necessary and sufficient conditions for a unique decomposition and synthesis of an r fuzzy set into the family of ordinary subsets p cuts are given as a consequence it is possible to obtain a fuzzy set as a synthesis of any collection of characteristics functions on a nonvoid set which was impossible in classical fuzzy set theory as a contribution to coding theory a possibility to express any binary block code syntheticly by a single fuzzy set is obtained note that up to now only some classes of binary codes have been characterized by fuzzy sets suitable examples bcd and a linear code are given at the end of the paper together with necessary and sufficient conditions under which an r valued fuzzy set corresponds to a linear code

lattice of partially ordered fuzzy subalgebras abstract fuzzy subalgebras are defined to be mappings from an algebra to a poset which can be a lattice and also the interval 0 1 so that every level subset is an ordinary subalgebra every fuzzy subalgebra has a canonical representation a mapping from the algebra to the poset of levels necessary and sufficient conditions under which a fuzzy subalgebra can be explicitly described by a suitable formula in lattice theoretic terms are given classes of algebras for which the poset of fuzzy subalgebras is a lattice are characterized

a note on the lattice of weak l valued congruences on the group of prime order abstract weak l valued fuzzy congruences on an algebra where l is a lattice were induced in 4 in this note we give a representation theorem for one subdirect power of l by means of the lattice of l valued congruences on any cyclic group of prime order

non standard cut classification of fuzzy sets abstract several important non standard cut sets of lattice valued fuzzy sets are investigated these are strong cuts not less and neither less nor equal cuts in each case it is proved that collection of all cuts of any lattice valued fuzzy set form a complete lattice under inclusion decomposition theorem representation by cuts is proved for neither less nor equal cuts necessary and sufficient conditions under which two lattice valued fuzzy sets with the same domain have equal families of corresponding cut sets are given

a note on cut worthiness of recognizable tree series the topic of this short note is tree series over semirings with partially ordered carrier set thus tree series become poset valued fuzzy sets starting with a collection of recognizable tree languages we construct a tree series recognizable over any semiring such that the carrier set is a poset generated by the collection in the framework of fuzzy structures the starting collection becomes a subset of the collection of the corresponding cut sets under some stricter conditions it is even equal to this collection we also partially solve an open problem which was posed in borchardt et al cut sets as recognizable tree languages fuzzy sets and systems 157 2006 1560 1571 namely we show that if f is a given tree series over a partially ordered locally finite semiring a then f is recognizable if and only if there are finitely many cut sets of f and every cut set is a recognizable tree language

poset valued convexities we analyze cut properties of lattice and poset valued functions fuzzy sets of n dimensional real variable with respect to convexity in the lattice valued case convexity of a fuzzy set is equivalent with the convexity of its standard cuts dealing with non standard cuts we show that unless the space of the values is a chain such statement in general does not hold we present analogue results in the more general setting of poset valued functions we also prove that there exist lattice and poset valued functions whose cuts are precisely the convex sets of particular families our work is motivated by possible applications of convex functions e g in image processing

representing ordered structures by fuzzy sets an overview we present a survey on representations of ordered structures by fuzzy sets any poset satisfying some finiteness condition semilattice lattice belonging to a special class e g distributive noetherian complete and others can be represented by a single function i e by a fuzzy set its domain and co domain are particular subsets of the same structure and consist of irreducible elements the representation is minimal in the sense that another representation could not be obtained by replacing the domain of the former by its proper subset by this approach the structure itself is uniquely represented by the collection of cuts ordered dually to inclusion

on the uniqueness of l fuzzy sets in the representation of families of sets abstract this paper deals with the uniqueness of l fuzzy sets in the representation of a given family of subsets of a nonempty set it first shows a formula of the number of l fuzzy sets whose collection of cuts coincides with a given family of subsets of a nonempty set and then provides a necessary and sufficient condition under which such l fuzzy set is unique

on lattice valued up sets and down sets isotone and anti isotone mappings from a poset into a complete lattice are investigated as lattice valued up sets and down sets respectively cuts of these are shown to be analogue crisp sub posets of the domain up set or semi filters and down sets or semi ideals the collection of all lattice valued up sets down sets of a poset is a complete lattice under the order inherited from the lattice among other results for a collection of crisp up sets down sets of a poset necessary and sufficient conditions are given under which this collection consists of cuts of a lattice valued up set down sets a generalization in the sense of closed fuzzy sets with respect to fuzzy relations is also carried out

rigidity of proper holomorphic self mappings of the pentablock abstract the pentablock is a hartogs domain in c 3 over the symmetrized bidisc in c 2 the domain is a bounded inhomogeneous pseudoconvex domain which does not have a c 1 boundary recently agler lykova young constructed a special subgroup of the group of holomorphic automorphisms of the pentablock and kosinski fully described the group of holomorphic automorphisms of the pentablock the aim of the present study is to prove that any proper holomorphic self mapping of the pentablock must be an automorphism

a generalization of the representation theorem abstract we generalize the representation theorem for fuzzy sets more specifically we give necessary and sufficient conditions for a family of sets to represent a permutation of the levels of some fuzzy set

algebraic fuzzy systems abstract a general theory of algebraic fuzzy systems is developed by introducing the notion of a fuzzy l subset corresponding to a set l of subsets of a nonempty set x assuming truth values in a complete brouwerian lattice it is observed that the set fl of all fuzzy l subsets of x is a closure algebraic closure fuzzy set system if and only if l is a closure algebraic closure set system on x a fuzzy l subset generated by a fuzzy subset is described modularity and distributivity of the complete lattices l and fl are discussed a method to construct fuzzy l subsets from certain family of members of l is established certain applications of the general theory are mentioned

a commutant lifting theorem for a domain in c2and spectral interpolation abstract we characterise the pairs of commuting operators on hilbert space for which the domain 1 2 1 2 1 1 2 1 is a complete spectral set we give an application to the spectral nevanlinna pick problem we obtain a necessary condition for the existence of an analytic 2 2 matrix function satisfying interpolation conditions and bounds on eigenvalues

spectral sets and distinguished varieties in the symmetrized bidisc abstract we show that for every pair of matrices s p having the closed symmetrized bidisc as a spectral set there is a one dimensional complex algebraic variety in such that for every matrix valued polynomial f z 1 z 2 f s p max z 1 z 2 f z 1 z 2 the variety is shown to have the determinantal representation s p det f p f s i 0 where f is the unique matrix of numerical radius not greater than 1 that satisfies s s p i p p 1 2 f i p p 1 2 when s p is a strict contraction then is a distinguished variety in the symmetrized bidisc i e a one dimensional algebraic variety that exits the symmetrized bidisc through its distinguished boundary we characterize all distinguished varieties of the symmetrized bidisc by a determinantal representation as above

uniqueness results in the representation of families of sets by fuzzy sets the important concept of identifying a fuzzy subset of a set r with its corresponding chain of level sets is widely used in the literature we investigate in this paper the problem of characterization of all fuzzy subsets of r that can be identified with a given arbitrary family c of subsets of r together with a given arbitrary subset s of 0 1 although our main focus is on the problem of uniqueness we start by giving necessary and sufficient conditions for both existence and uniqueness of such fuzzy subsets we continue by obtaining several algebraic and topological properties under which uniqueness is guaranteed to further support our results we present some examples which we hope will shed more light on the above mentioned problem of identification we finish by raising a few natural questions that are left as open problems for further investigation

fuzzy groups and collections of subgroups abstract the aim of the paper is to investigate fuzzy subgroups of a group from a general point of view using collections of ordinary subgroups fuzzy subgroups are taken to be special mappings from a group to a partially ordered set which can also be a lattice and particularly the unit interval 0 1 as it is known every fuzzy subgroup uniquely determines a poset of level subgroups of the same group properties of that poset are investigated on the other hand it is proved that every collection of subgroups can be used for the construction of a suitable fuzzy subgroup of a group for that purpose particular fuzzy completions of collections of subsets are introduced it turns out that fuzzy subgroup is an intrinsic notion expressible in terms of collections of subgroups under set theoretic relations and operations

representation of lattices by fuzzy sets abstract we prove that every lattice l of finite length can be represented by a fuzzy set on the collection x of meet irreducible elements of l a decomposition of this fuzzy set gives a family of isotone functions from x to 2 0 1 the lattice of which is isomorphic to l more generally conditions under which any collection of isotone functions from a finite set into 2 corresponds to a decomposition of a fuzzy set are given as a consequence the representation theorem for a finite distributive lattice by the lattice of all isotone functions is obtained the collection of all lattices characterized by the same fuzzy set turns out to be a lattice with the above mentioned distributive lattice as the greatest element

homomorphisms of poset valued algebras the notion of a homomorphism for fuzzy algebraic structures having arbitrary possibly different partially ordered sets or lattices as co domains is introduced basic properties of homomorphisms embeddings and isomorphisms are investigated and fundamental homomorphism and isomorphism theorems are proved

from fuzzy sets to the decompositions of non rigid sets the concept of identifying a fuzzy subset of a set r with its collection of level sets and its collection of membership values plays an important role in many real life applications such identification is possible only if the fuzzy set can be recaptured this implies that uniqueness of the fuzzy set plays an important role it has been recently shown that in order to have uniqueness it is necessary and sufficient that the collection s of membership values of the fuzzy set be a rigid set with this in mind the authors went on to show that if s has the min max property then s is a rigid set and left the converse as an open problem we start this paper by constructing a counterexample that shows that rigid sets do not have to have the min max property we then take a closer look at non rigid sets by decomposing them into isomorphism invariant components and into s connected components in the hope of getting a characterization of non rigid sets which will lead to a characterization of rigid sets and hence of uniqueness of the fuzzy set we also investigate in the case of nonuniqueness the number of fuzzy sets corresponding to a collection of level sets and a collection of membership values

the lattice of l ideals of a ring is modular in this paper we extend the notion of a tip extended pair of fuzzy subgroups to l ideals of a ring we prove that the sum of two tip extended l ideals of an arbitrary pair of l ideals of a ring is the least l ideal containing the union of the given l ideals using this construction of join of l ideals we prove that the lattice of all l ideals of a given ring is modular

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

the complex geometry of a domain related to μ synthesis abstract we establish the basic complex geometry and function theory of the pentablock p which is the bounded domain p a 21 tr a det a a a i j i j 1 2 b where b denotes the open unit ball in the space of 2 2 complex matrices we prove several characterisations of the domain we show that p arises naturally in connection with a certain robust stabilisation problem in control theory the problem of synthesis we describe the distinguished boundary of p and exhibit a 4 parameter group of automorphisms of p we demonstrate connections between the function theories of p and b we show that p is polynomially convex and starlike and we show that the real pentablock p r 3 is a convex set bounded by five faces three of them flat and two curved

completion of ordered structures by cuts of fuzzy sets an overview the aim of the paper is to present a role of fuzzy sets in the theory of ordered structures main algebraic properties of cuts of fuzzy sets are given and a completion of partially ordered sets to complete lattices is described it turns out that this completion is equivalent with the famous dedekind macneille completion but the algorithm presented here is much simpler

l fuzzy sets and codes abstract a decomposition of an l valued finite fuzzy set l is a lattice gives a family of characteristic functions which can be considered as a binary block code using a previous theorem of synthesis for fuzzy sets we give conditions under which an arbitrary block code corresponds to an l valued fuzzy set an explicit description of the hamming distance as well as of any code distance is also given all in lattice theoretic terms finally we give necessary and sufficient conditions under which a linear code corresponds to an l valued fuzzy set it turns out that in such case the lattice l has to be boolean

on a generalization of fuzzy algebras and congruences abstract partially ordered fuzzy algebras are mappings from an algebra to a partially ordered set with the property that every level subset is an ordinary subalgebra similar definitions are induced for p valued congruences and weak congruences necessary and sufficient conditions under which an arbitrary collection of subalgebras congruences enables construction of a p valued fuzzy subalgebra congruence are given any p valued weak congruence uniquely determines a p valued subalgebra of the same algebra finally any collection of subalgebras or congruences of a given algebra can be used for the construction of a relational valued fuzzy algebra or congruence this seems to be the most general way to obtain a fuzzy algebra congruence out of the collection of the ordinary subalgebras congruences

short communication lattice representations of interval valued fuzzy sets in this paper a solution of the problem of the interval valued fuzzy sets synthesis is given the codomain is considered to be a lattice of intervals the necessary and sufficient conditions for a family of subsets to be a family of cut sets of an interval valued fuzzy set are given

uniqueness in the generalized representation by fuzzy sets the concept of representation of a given family of sets by a fuzzy set in such a way that the level sets of the fuzzy set are precisely the given sets is widely used in the literature for the classification and study of fuzzy algebraic structures on the same underlying set the problem of existence of such fuzzy sets was investigated by various authors as far as we know the problem of uniqueness of such fuzzy sets was not studied before we establish among other things necessary and sufficient conditions under which uniqueness is guaranteed in the cases of complete as well as partial representations of the level sets of the fuzzy set

dilations of γ contractions by solving operator equations for a contraction p and a bounded commutant s of p we seek a solution x of the operator equation s s p 1 p p 1 2 x 1 p p 1 2 where x is a bounded operator on ran over bar 1 p p 1 2 with numerical radius of x being not greater than 1 a pair of bounded operators s p which has the domain gamma z 1 z 2 z 2 vertical bar z 1 vertical bar 1 vertical bar z 2 vertical bar 1 subset of c 2 as a spectral set is called a p contraction in the literature we show the existence and uniqueness of solution to the operator equation above for a gamma contraction s p this allows us to construct an explicit gamma isometric dilation of a gamma contraction s p we prove the other way too i e for a commuting pair s p with parallel to p parallel to 1 and the spectral radius of s being not greater than 2 the existence of a solution to the above equation implies that s p is a gamma contraction we show that for a pure f contraction s p there is a bounded operator c with numerical radius not greater than 1 such that s c c p any gamma isometry can be written in this form where p now is an isometry commuting with c and c any gamma unitary is of this form as well with p and c being commuting unitaries examples of gamma contractions on reproducing kernel hilbert spaces and their gamma isometric dilations are discussed c 2012 elsevier inc all rights reserved

on a representation of posets by fuzzy sets the aim of the paper is to present an application of fuzzy sets in mathematics namely in the theory of ordered sets an algorithm for the construction of p fuzzy sets with distinct levels is given in connection with this every finite poset can be mapped by an anti isotone bijection onto the poset of levels of a suitable fuzzy set moreover it is proved that every finite partially ordered set p can be represented by the poset of levels of a particular fuzzy set defined on the collection of meet irreducible elements of p

cuts of poset valued functions in the framework of residuated maps abstract we analyze cuts of poset valued functions relating them to residuated mappings dealing with the lattice valued case we prove that a function x l induces a residuated map f l p x whose values are the cuts of and we describe the corresponding residual conversely it turns out that every residuated map f from l to the power set of x determines a lattice valued function x l whose cuts coincide with the values of f for general poset valued functions we give conditions under which the map sending an element of a poset to the corresponding cut is quasi residuated and then conditions under which it is also residuated we prove that without additional conditions the map analogue to the residual is a partial function hence we get particular weakly residuated maps which on the power set of the domain generate centralized systems instead of closures we show that the main properties of residuated maps are preserved in this generalized case we apply these results to the canonical representation of poset valued and lattice valued functions using the corresponding closures and centralized systems