on fuzzy spaces and fuzzy group theory abstract the concept of fuzzy space is introduced this concept corresponds to the concept of the universal set in the ordinary case the algebra of fuzzy spaces and fuzzy subspaces is studied using the concept of fuzzy space and fuzzy binary operation a new approach to study fuzzy group theory is given this approach can be considered as a generalization and reformulation of the rosenfeld theory of fuzzy subgroups therefore it is an active tool to develop the theory of fuzzy groups

a note on fuzzy relations and fuzzy groups abstract it is shown that for a group g if is a fuzzy subgroup of g for all fuzzy subgroups of g g then either g is a cyclic group of order p m for some m 1 or g z p where p is a prime conversely we prove that if either g is a cyclic group of order p m for some m 1 or g z p then is a fuzzy subgroup of g for any fuzzy subgroup of g g where p is a prime

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

fuzzy groups some group theoretic analogs abstract we introduce the concepts of 1 order of a fuzzy subgroup of a finite group 2 a fuzzy abelian group and 3 a fuzzy solvable group we obtain some results involving these notions which are analogs of results from group theory also we prove analogs of two basic theorems of group theory namely cayley s theorem and lagrange s theorem

a contribution to the theory of fuzzy subgroups abstract in this paper we continue our study in order to give a partial answer to the question suggested by asaad in 1991 what can be said about the structure of a group g when the information are known about the structure of a fuzzy subgroup a of g and viceversa

the free product of fuzzy subgroups abstract an associative product to be called the free product of fuzzy sets in a semigroup is defined which lies between sup min and sup max products it behaves nicely for characteristic functions and fuzzy point sets and simulates all the standard results for the product of two subgroups of a group instead of fuzzy invariance which is too strong only a milder form of relative fuzzy invariance is used to establish fuzzy subgroupness of the product which is intimately connected with the necessity of two fuzzy subgroups to commute lastly under a slightly restricted situation the free product induces a fuzzy subgroup on a quotient group it is suggested that the product be tested for a suitable definition of a fuzzy topological group an analysis of its deeper aspects is forthwith

on t fuzzy subfield and t fuzzy vector subspaces abstract in this paper we improve the definition of t fuzzy subfield and observe some of its properties the concept of t fuzzy vector subspaces of a t fuzzy subfield of a field will also be investigated and we further introduce the concept of t fuzzy subalgebra

representation of lattices by fuzzy weak congruence relations fuzzy lattice valued weak congruences of abstract algebras are investigated for an algebra the family of all such fuzzy relations is a complete lattice its structure and cut properties are investigated and fully described these fuzzy weak congruences are applied in representation of complete and algebraic lattices a wider class of lattices can be represented in such a fuzzy framework than in classical algebra we prove that there is a straightforward representation of any complete lattice using it as a co domain in a more general case it is proved that several subdirect powers of lattices are also representable by fuzzy weak congruences

groups and fuzzy subgroups abstract the purpose of this paper is to deal with the following two problems 1 what can be said about the structure of a fuzzy subgroup a of a group g when information is known about the structure of g 2 what can be said about the structure of a group g when information is known about the structure of a fuzzy subgroup a of g

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

on generalized characteristic fuzzy subgroups of a finite group abstract in 1991 asaad 2 suggested the following two questions i what can be said about the structure of a fuzzy subgroup a of a finite group g when information is known about the structure of g ii what can be said about the structure of group g when information is known about the structure of a fuzzy subgroup a of g the purpose of this paper is to continue the investigation of the above questions

fuzzy cosets and fuzzy normal subgroups abstract in this paper besides generalizing some of the results of mukherjee and bhattacharya 9 we investigate conditions under which a homomorphic image of a fuzzy normal subgroup is fuzzy normal we also prove that a fuzzy subgroup of order p n 1 of a group of order p n is a fuzzy normal subgroup if p is a prime

fuzzy subgroups having the property abstract we introduce the notion of the property for a fuzzy group and characterize all finite cyclic groups in terms of this notion this property concerns the transition of the order of an element of a group from the classical to the fuzzy setting

fuzzy partial groups abstract the fuzzification of partial monoids and partial groups has introduced in a little different way than that one usually found the notion of the t partial monoid on i has been investigated and is used to define different membership values for the partial identities which is consistent with the hall spirit of the fuzzification this is in fact the motivation of this paper examples have been given to get the generalization reasonable in fact the basic results of fuzzy groups and fuzzy monoids has been generalised

fuzzy identities with application to fuzzy semigroups the notion of a fuzzy identity is introduced on a fuzzy subalgebra in the framework of fuzzy equalities instead of the crisp one a fuzzy identity may be satisfied by a fuzzy subalgebra while the underlying crisp algebra need not satisfy the analogous crisp identity among other properties it is shown that for every fuzzy subalgebra m of a there is a least fuzzy equality such that a fuzzy identity holds on m as an application fuzzy semigroups are introduced with respect to a fuzzy equality in particular fuzzy semilattices cancellative and regular fuzzy semigroups are investigated

numerical methods for second order singular pertubation problems abstract by using sufficient conditions which ensure the well conditioning of tridiagonal matrices we derive numerical methods for second order singular pertubations boundary values problems under hypotheses similar to the ones usually used in the continuous theory methods are derived which use constant step sizes h o 1 2 in the more general case methods with variable step sizes are defined which solve the problem with n 1 steps examples on standard test problems are shown

on the definition of a fuzzy subgroup abstract using the idea of quasi coincidence of a fuzzy point with a fuzzy set some new concepts of a fuzzy subgroup are introduced and their acceptibility is investigated some fundamental properties of one such viable fuzzy subgroup are obtained

e fuzzy groups an e fuzzy group is a lattice valued algebraic structure defined on a crisp algebra which is not necessarily a group the crisp equality is replaced by a particular fuzzy one denoted by e the classical group like properties are formulated as appropriate fuzzy identities special lattice theoretic formulas we prove basic features of e fuzzy groups properties of the unit and inverses cancellability solvability of equations subgroup properties and others we also prove that for every cut of an e fuzzy group which is a classical subalgebra of the underlying algebra the quotient structure over the corresponding cut of the fuzzy equality is a classical group

fuzzy cosets and cyclic and abelian fuzzy subgroups abstract here we give some properties of fuzzy cosets normal fuzzy subgroups and s fuzzy subgroups also we introduce the notions of cyclic abelian fuzzy subgroups and prove some of their properties

fuzzy hypergroups based on fuzzy relations based on fuzzy reasoning in fuzzy logic this paper studies a fuzzy hyperoperation and a fuzzy hypergroupoid associated with a fuzzy relation a sufficient and necessary condition for such a fuzzy hypergroupoid being a fuzzy hypergroup is given and the properties of the fuzzy hypergroups associated with fuzzy relations are investigated furthermore the definition of normal fuzzy hypergroups is put forward and it is shown that the category nfhg of normal fuzzy hypergroups satisfies all the axioms of topos except for the subobject classifier axiom

fuzzy semiprimary ideals of rings abstract the concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings this class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals it is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary a study of i the level ideals of a fuzzy semiprimary ideal ii the characteristic function of a semiprimary ideal iii the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and iv the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms is carried out

fuzzy irreducible ideals in rings abstract in this paper the concept of a fuzzy set is extended by introducing the concept of a fuzzy irreducible ideal in a ring the characteristic function of an irreducible ideal and the level ideals of a fuzzy irreducible ideal are studied the algebraic nature of homomorphic images and preimages of fuzzy irreducible ideals is also examined some other fundamental results pertaining to this concept are discussed

level subgroups and union of fuzzy subgroups abstract in this paper we investigate the conditions under which a given fuzzy subgroup of a given group can or can not be realized as a union of two proper fuzzy subgroups moreover we formulate the notion of fuzzy subgroup generated by a given fuzzy subset by level subgroups we also give characterizations of fuzzy conjugate subgroups and fuzzy characteristic subgroups by their level subgroups also we discuss the relationship between level subgroups of a given fuzzy subgroup and the level subgroups of its homomorphic image

fuzzy subgroups and anti fuzzy subgroups abstract we introduce the concept of anti fuzzy subgroups of groups lower level subset of a fuzzy subset lower level subgroups and prove some results on these we show that a fuzzy subset of a group is a fuzzy subgroup iff the complement of this fuzzy subset is anti fuzzy subgroup we fuzzify lower level subsets and prove that if a fuzzy set is an anti fuzzy subgroup then the fuzzifications of its lower level subsets are also anti fuzzy subgroups

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

finitely fuzzy value distribution of fuzzy vector spaces and fuzzy modules abstract we discuss the problem of finitely fuzzy value distribution of fuzzy vector spaces and fuzzy modules a sufficient and necessary condition is obtained in this paper

isomorphic fuzzy groups abstract we introduce an acceptable definition of isomorphic fuzzy groups which fulfils the basic requirements the fuzzy versions of the first and the second isomorphism theorems become immediate consequences the idea is applied to the sup min product of two fuzzy subgroups of a group as an indispensable tool the characterization of a fuzzy normal subgroup of a group by the so called open level subgroups is brought forth an isomorphism between the cartesian product and the sup min product of two fuzzy subgroups of a group is demonstrated

free q schrodinger equation from homogeneous spaces of the 2 dim euclidean quantum group after a preliminary review of the definition and the general properties of the homogeneous spaces of quantum groups the quantum hyperboloidqh and the quantum planeqp are determined as homogeneous spaces offq e 2 the canonical action ofeq 2 is used to define a naturalq analog of the free schrodinger equation that is studied in the momentum and angular momentum bases in the first case the eigenfunctions are factorized in terms of products of twoq exponentials in the second case we determine the eigenstates of the unitary representation which in theqp case are given in terms of hahn exton functions introducing the universalt matrix foreq 2 we prove that the hahn exton as well as jacksonq bessel functions are also obtained as matrix elements oft thus giving the correct extension to quantum groups of well known methods in harmonic analysis

basic fuzzy subgroups abstract the concepts of p independence and basic subgroups for abelian groups are introduced for fuzzy subgroups the notions of basic fuzzy subgroups and direct sums of fuzzy subgroups are used to determine a system of invariants for certain fuzzy subgroups

galois theory for fuzzy fields abstract the aim of this paper is to present a generalized form of the galois connection between the intermediate fields of a field extension and the subgroups of the corresponding galois group we prove that the fundamental theorem of galois theory accepts a generalization in the fuzzy framework

more on fuzzy lattices in this paper the concept of an q k fuzzy sublattice ideal filter and q k fuzzy sublattice ideal filter in a lattice is introduced which is a generalization of the concept as given by davvaz and kazanci 2011 18 characterizations for an q k fuzzy sublattice ideal filter and q k fuzzy sublattice ideal filter are provided different classes of lattices are characterized by the properties of these fuzzy sublattices ideals filters using the notions of a fuzzy sublattice ideal filter with thresholds characterization of a fuzzy sublattice ideal filter an q k fuzzy sublattice ideal filter and q k fuzzy sublattice ideal filter are discussed

fuzzy commutative ideals of bck algebras abstract in this paper we define and discuss the fuzzy commutative ideals in bck algebras a fuzzy characteristic commutative ideal is characterized in terms of its level commutative ideals

structure and construction of fuzzy subgroups of a group abstract we define the order of a fuzzy set and give conditions for different order fuzzy sets to be fuzzy subgroups through their level subgroups finally we give a list of fuzzy subgroups of order 1 6