an efficient ml decoder for tail biting codes based on circular trap detection tail biting codes are efficient coding techniques to eliminate the rate loss in conventional known tail convolutional codes at a cost of increased complexity in decoders in addition tail biting trellis representation of block codes makes the trellis based maximum likelihood ml decoder desirable for implementation circular viterbi algorithm cva is introduced to decode the tail biting codes for its decoding efficiency however its decoding process suffers from circular traps which degrade the decoding efficiency in this paper we propose an efficient checking rule for the detection of circular traps based on this rule a novel maximum likelihood ml decoding algorithm for tail biting codes is presented on tail biting trellis computational complexity and memory consumption of this decoder are significantly reduced comparing to other available ml decoders such as the two phase ml decoder to further reduce the decoding complexity we propose a new near optimal decoding algorithm based on a simplified trap detection strategy the performance of the above algorithms is validated with simulation

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

capacity and codes for embedding information in gray scale signals gray scale signals can be represented as sequences of integer valued symbols if such a symbol has alphabet 0 1 2 sup b 1 it can be represented by b binary digits to embed information in these sequences we are allowed to distort the symbols the distortion measure that we consider here is squared error however errors larger than m are not allowed the embedded message must be recoverable with error probability zero in this setup there is a so called rate distortion function that tells us what the largest embedding rate is given a certain distortion level and parameter m first we determine this rate distortion function for m 1 and for m spl rarr spl infin next we compare the performance of low bits modulation to the rate distortion function for m spl rarr spl infin then embedding codes are proposed based on i ternary hamming codes and on the ii ternary golay code we show that all these codes are optimal in the sense that they achieve the smallest possible distortion at a given rate for fixed block length for any m

tailbiting codes bounds and search results tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied the notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower bounded the asymptotic behavior of the derived bound is investigated furthermore for regular tailbiting codes the product state complexity is lower bounded tables of new tailbiting trellis representations of linear block codes of rates 1 2 1 3 and 1 4 are presented almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time varying trellises which are optimal five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters four of them are also superior to any previously known nonlinear code with the same parameters also more than 40 other quasi cyclic codes have been found that improve the parameter set of previously known quasi cyclic codes

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 the efficiency of embedding information within scalar quantizers this correspondence explores the efficiency of embedding information within scalar quantizers the aim being to efficiently include additional information within an existing coding scheme two different information embedding problems are considered one using post quantization embedding pqe and the other using joint quantization embedding high resolution bounds are derived for the achievable embedding rates in both these problems surprisingly these results show that it is possible to ldquosoak uprdquo the inefficiencies in scalar quantization through the embedding process the correspondence concludes by considering some practical examples of low embedding rate coding strategies

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

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

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

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

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

on an equivalence of fuzzy subgroups ii this paper forms a sequel to the paper on an equivalence of fuzzy subgroups i here we determine the number of distinct equivalence classes of fuzzy subgroups of g z p 1 z p n where p 1 p 2 p n are distinct primes we introduce the notion of a keychain of a chain of length n 1 and index of a keychain in order to determine the number of fuzzy subgroups of g we achieve this by using induction on the index of a keychain

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 new representation theorem of l subsets abstract in the paper the implication and tensor operations of strong l nested systems are found and further we show that the family of all strong l nested systems on a universal set forms a complete girard residuated lattice provided the underlying complete residuated lattice is girard finally an isomorphism of complete girard residuated lattices between the family of all strong l nested systems and the l power set for any fixed universal set is established and thus a new representation theorem of l subsets is obtained

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

t norm based cuts of intuitionistic fuzzy sets abstract we consider the system of intuitionistic fuzzy sets if sets in a universe x and study the cuts of an if set suppose a left continuous triangular norm is given the t norm based cut level set of an if set is defined in a way that binds the membership and nonmembership functions via the triangular norm this is an extension of usual cuts of if sets we show that the system of these cuts fulfils analogical properties as usual systems of cuts however it is not possible to reconstruct an if set from the system of t norm based cuts

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

the extension principle and a decomposition of fuzzy sets we give an algorithm to decompose a fuzzy interval u using this decomposition and the multilinearization of a univariate function f we obtain an approximation of the fuzzy interval where is obtained from f by applying the extension principle we provide approximation bounds some numeric illustration is provided

decomposition and construction of fuzzy sets and their applications to the arithmetic operations on fuzzy quantities abstract the membership function of a fuzzy set can be constructed from a family of subsets of a universal set via the form of resolution identity assuming that the family is nested in this paper we are going to consider the weak case of assuming that only a countable subfamily is nested using this generalized consideration we can overcome the ambiguity of arithmetic operations on fuzzy quantities two well known additions of fuzzy quantities in vector space have been adopted in the literature one is based on the extension principle by directly considering the membership functions of course this addition always exists another one uses the level sets without considering the membership functions however the existence of this addition cannot be guaranteed in this paper we are going to propose two other additions of fuzzy quantities in vector space one generalizes the extension principle and the other one also uses the level sets without considering the membership functions such that this generalized addition always exists we can show that these four additions are equivalent

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

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 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

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

tight bounds on the minimum average weight per branch for rate n 1 n convolutional codes consider a cycle in the state diagram of a convolutional code the average weight per branch of the cycle is equal to the total hamming weight of all labels on the branches divided by the number of branches let w sub 0 be the minimum average weight per branch over all cycles in the state diagram except the zero state self loop of weight zero codes with low w sub 0 result in high bit error probabilities when they are used with either viterbi or sequential decoding hemmati and costello 1980 showed that w sub 0 is upper bounded by 2 sup spl nu 2 3 spl middot 2 sup spl nu 2 1 for a class of 2 1 codes where spl nu denotes the constraint length in the present correspondence it is shown that the bound is valid for a large class of n n 1 codes n spl ges 2 examples of high rate codes with w sub 0 equal to the upper bound are also given hemmati and costello defined a class of codes to be asymptotically catastrophic if w sub 0 approaches zero for large spl nu the class of n n 1 codes constructed by wyner and ash 1963 is shown to be asymptotically catastrophic all codes in the class have minimum possible w sub 0 equal to 1 spl nu

tailbiting codes obtained via convolutional codes with large active distance slopes the slope of the active distances is an important parameter when investigating the error correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes the slope of the active distances is equal to the minimum average weight cycle in the state transition diagram of the encoder a general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived moreover a search technique resulting in new tables of rate r 1 2 and rate r 1 3 convolutional encoders with high memories and large active distance slopes is presented furthermore we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances tables of rate r 1 2 and rate r 1 3 tailbiting codes with larger minimum distances than the best previously known quasi cyclic codes are given two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length one of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters

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

a beast for prowling in trees when searching for convolutional codes and tailbiting codes of high complexity it is of vital importance to use fast algorithms for computing their weight spectra which corresponds to finding low weight paths in their code trellises this can be efficiently done by a combined search in both forward and backward code trees a bidirectional efficient algorithm for searching such code trees beast is presented for large encoder memories it is shown that beast is significantly more efficient than comparable algorithms beast made it possible to find new convolutional and tailbiting codes that have larger free minimum distances than the previously best known codes with the same parameters tables of such codes are presented

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

on joint information embedding and lossy compression we consider the problem of optimum joint information embedding and lossy compression with respect to a fidelity criterion the goal is to find the minimum achievable compression composite rate r sub c as a function of the embedding rate r sub e and the average distortion level spl delta allowed such that the average probability of error in decoding of the embedded message can be made arbitrarily small for sufficiently large block length we characterize the minimum achievable composite rate for both the public and the private versions of the problem and demonstrate how this minimum can be approached in principle we also provide an alternative single letter expression of the maximum achievable embedding rate embedding capacity as a function of r sub c and spl delta above which there exist no reliable embedding schemes

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

preferential normal fuzzy subgroups in this paper we consider the notion of normality on preferential fuzzy subgroups of a finite group g and define the associated concept of a normal pinned flag we discuss the preferential equality of quotients and products of normal preferential fuzzy subgroups further normalizers of fuzzy subgroups under preferential equality are briefly dealt with examples are given to illustrate the structure of preferential normal fuzzy subgroups and normal pinned flags

on existence of p valued fuzzy sets with a given collection of cuts a representation theorem of families of subsets by poset valued fuzzy sets is presented namely necessary and sufficient conditions are given under which for a given family of subsets f of a set x and a fixed poset p there is a fuzzy set m x p such that the collection of cuts of m coincides with f

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

on the construction of adjunctions between a fuzzy preposet and an unstructured set in this work we focus on adjunctions also called isotone galois connections in the framework of fuzzy preordered sets hereafter fuzzy preposets specifically we present necessary and sufficient conditions so that given a mapping f a b from a fuzzy preposet a into an unstructured set b it is possible to construct a suitable fuzzy preorder relation on b for which there exists a mapping g b a such that the pair f g constitutes an adjunction

some new binary quasi cyclic codes by means of local search techniques five quasi cyclic codes have been found that have a higher minimum distance than known binary linear codes the new codes have parameters 102 17 37 60 20 17 84 21 27 105 21 36 and 100 25 30 also 39 other quasi cyclic codes have been found that improve the parameter sets of previously known quasi cyclic codes twenty four of them give a new and easier description of binary linear codes with best known parameters

active distances for convolutional codes a family of active distance measures for general convolutional codes is defined these distances are generalizations of the extended distances introduced by thommesen and justesen 1983 for unit memory convolutional codes it is shown that the error correcting capability of a convolutional code is determined by the active distances the ensemble of periodically time varying convolutional codes is defined and lower bounds on the active distances are derived for this ensemble the active distances are very useful in the analysis of concatenated convolutional encoders

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

woven convolutional codes i encoder properties encoders for convolutional codes with large free distances can be constructed by combining several less powerful convolutional encoders this paper is devoted to constructions in which the constituent convolutional codes are woven together in a manner that resembles the structure of a fabric the general construction is called twill and it is described together with two special cases viz woven convolutional encoders with outer warp and with inner warp the woven convolutional encoders inherit many of their structural properties such as minimality and catastrophicity from their constituent encoders for all three types of woven convolutional codes upper and lower bounds on their free distances as well as lower bounds on the active distances of their encoders are derived

some best rate 1 p and rate p 1 p systematic quasi cyclic codes tables of 1 p and rate p 1 p binary quasi cyclic qc codes that extend previously published results are presented many of these codes attain bounds given by t verhoeff 1987 who composed a table of bounds on the maximum possible minimum distance of binary linear codes many of the codes presented meet or improve these bounds a best code is considered to be one which has the largest possible minimum distance for the given code dimensions n and k and class of error correcting codes a good code has the maximum known minimum distance for the class of codes an optimal code is one that achieves the maximum possible minimum distance for a linear code with the same dimensions binary power residue codes are found and used to construct qc codes

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

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

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

a note on a natural equivalence relation on fuzzy power set the necessary and sufficient conditions under which two fuzzy sets with the same domain have equal families of cut sets are given consequently there is a corresponding equivalence relation on the related fuzzy power set the collection of classes under this relation can be ordered and we give a necessary and sufficient condition under which it is a lattice

a note on asymptotically catastrophic convolutional codes of rate n 1 n convolutional codes with rate r n 1 n n spl ges 2 are defined in terms of their minimal parity check matrices the matrices are represented by a binary vector notation introduced by ytrehus 1992 the upper bounds on spl omega 0 minimum average weight per branch are presented many classes of rate n 1 n convolutional codes are shown to be asymptotically catastrophic