extensions of fuzzy subrings and fuzzy ideals abstract in this paper we give necessary and sufficient conditions for a fuzzy subring or a fuzzy ideal a of a commutative ring r to be extended to one a e of a commutative ring s containing r as a subring such that a and a e have the same image one of the applications of these results gives a criterion for a fuzzy subring of an integral domain r to be extendable to a fuzzy subfield of the quotient field of r

tl subrings and tl ideals part 4 radicals of tl ideals abstract in this paper the notion of the radicals of l subsets and tl ideals of a ring is introduced and studied where t is an infinitely distributive t norm on a given complete brouwerian lattice l our main objective is to present basic properties about the radicals of tl ideals in preparation for the forthcoming two papers

rough set theory applied to fuzzy ideal theory we use covers of the universal set to define approximation operators on the power set of the given set in section 1 we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets we define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the fuzzy prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of fuzzy ideals of a ring

time analyticity and backward uniqueness of weak solutions of the navier stokes equations of multidimensional compressible flow abstract we prove that solutions of the navier stokes equations of three dimensional compressible flow restricted to fluid particle trajectories can be extended as analytic functions of complex time as consequences we derive backward uniqueness of solutions as well as sharp rates of smoothing for higher order lagrangean time derivatives the solutions under consideration are in a reasonably broad regularity class corresponding to small energy initial data with a small degree of regularity the latter being required for conversion to the lagrangean coordinate system in which the analysis is carried out

fuzzy radicals and prime fuzzy ideals of ordered semigroups let s be an ordered semigroup a fuzzy subset of s is by definition an arbitrary mapping f s 0 1 where 0 1 is the usual interval of real numbers motivated by studying prime fuzzy ideals in rings semigroups and ordered semigroups and as a continuation of kehayopulu and tsingelis s works in ordered semigroups in terms of fuzzy subsets in this paper we introduce the notion of ordered fuzzy points of an ordered semigroup s and give a characterization of prime fuzzy ideals of an ordered semigroup s we also introduce the concepts of weakly prime fuzzy ideals completely prime fuzzy ideals completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of an ordered semigroup s and establish the relationship between the five classes of ideals furthermore we characterize weakly prime fuzzy ideals completely semiprime fuzzy ideals and weakly completely prime fuzzy ideals of s by their level ideals finally we introduce and investigate the fuzzy radicals of ordered semigroups by means of ordered fuzzy points and prove that the fuzzy radical of every completely semiprime fuzzy ideal of an ordered semigroup s can be expressed as the intersection of all weakly completely prime fuzzy ideals containing it as an application of the results of this paper the corresponding results of semigroups without order are also obtained

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

global smooth solutions for a class of parabolic integrodifferential equations the existence and uniqueness of smooth global large data solutions of a class of quasilinear partial integrodifferential equations in one space and one time dimension are proved if the integral kernel behaves like t near t 0 o with a 2 3 an existence and regularity theorem for linear equations with variable coefficients that are related to this type is also proved in arbitrary space dimensions and with no restrictions for o

fuzzy ideals and fuzzy semiprime ideals some ring theoretic analogues abstract an attempt is made to examine the algebraic nature of fuzzy semiprime ideals under homomorphisms also the fuzzy nil radical of the nth power of a fuzzy prime ideal is studied and a sufficient condition for a fuzzy ideal to be fuzzy maximal is given

on fuzzy ideals and fuzzy quotient rings of a ring abstract in this paper we define fuzzy quotient rings induced by fuzzy ideals and give the notion of fuzzy nilpotent ideals we give also a characterization of semiprime rings in terms of fuzzy ideals

tl subrings and tl ideals part 1 basic concepts abstract in 25 27 we put forward a rudimentary frame of the theory of tl subalgebras in 26 we established a theory of tl subgroups and tl subgroups in this paper we will continue our work in concern with basic ideas provided in 25 27 introduce the notions of tl subrings and tl ideals of a ring and study some of their properties where t is an arbitrary infinitely distributive t norm on a given complete brouwerian lattice l

some results on fuzzy prime spectrum of a ring abstract in this note we extend the results of kumar 1992 in this context we first give a counter example to his theorem 3 4 ii and then we prove that the fuzzy prime spectrum of a ring and the elements of its basis are both compact finally by giving two examples we will show that it is not true in general that any element of a basis of the fuzzy prime spectrum of a boolean ring is closed and the fuzzy prime spectrum itself is hausdorff also by an example it is shown that the proof of the necessity of the condition of theorem 5 3 of kumar 1992 is incorrect

on fuzzy quotient rings induced by fuzzy ideals abstract in this paper we define fuzzy quotient rings induced by fuzzy ideals and study the relation between fuzzy quotient rings and fuzzy ideals

t fuzzy prime ideals abstract the purpose of this paper is to define study and characterize the t fuzzy prime ideals t fuzzy strongly prime ideals and t s fuzzy weakly prime ideals proving the inclusion relationships that are satisfied among them the t fuzzy comaximal ideals and fuzzy maximal ideals are also investigated

fuzzy relations on rings and groups abstract let s be any nonempty set a fuzzy relation on s is a fuzzy subset of s s in this paper we study fuzzy relations on rings and groups in particular we show that if and are fuzzy left right ideals of a ring r then is a fuzzy left right ideal of r r and conversely if is a fuzzy left right ideal of r r then either or is a fuzzy left right ideal of r an example is given to show that if is a fuzzy left right ideal of r r then and both need not be fuzzy left right ideals of r an example is also given to show that if is a fuzzy left right ideal of r r then the weakest fuzzy subset of r on which is a fuzzy relation need not be a fuzzy left right ideal of r we obtain similar results for groups we also show that certain results of bhattacharya and mukherjee 1985 are not true

fuzzy subgroups fuzzy ideals and fuzzy cosets some properties abstract as an application of fuzzy subgroups a necessary and sufficient condition for a p group to be cyclic is obtained a procedure to construct the fuzzy subgroup generated by a fuzzy subset of a group is provided also the effect of a homomorphism on the sum product intersection of two fuzzy ideals is studied some elementary properties of the ring fuzzy cosets of a fuzzy ideal are also obtained

fuzzy primary representations of fuzzy ideals abstract let r be a commutative ring with identity and let a be a fuzzy ideal of r then a is said to have a fuzzy primary representation if a is the intersection of a finite number of fuzzy primary ideals we show that every fuzzy ideal a of r such that a 0 1 has a fuzzy primary representation if and only if r is artinian we show that uniqueness properties for reduced primary representations of ideals in the usual sense carry over to fuzzy ideals

regular fuzzy duo rings in this paper we shall introduce the concept of a fuzzy bi ideal a fuzzy quasi ideal in a ring and give some properties of such fuzzy ideals we also give a characterization of a regular fuzzy duo ring in terms of fuzzy ideals

roughness based on fuzzy ideals the theory of rough set proposed by pawlak and the theory of fuzzy set proposed by zadeh are complementary generalizations of classical set theory many sets are naturally endowed with two binary operations addition and multiplication one concept which does this is a ring this paper concerns a relationship between rough sets fuzzy sets and ring theory it is a continuation of ideas presented by kuroki and wang n kuroki p p wang the lower and upper approximations in a fuzzy group inform sci 90 1996 203 220 we consider a ring as a universal set and we assume that the knowledge about objects is restricted by a fuzzy ideal in fact we apply the notion of fuzzy ideal of a ring for definitions of the lower and upper approximations in a ring some characterizations of the above approximations are made and some examples are presented

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

universal variable to fixed length source codes a universal variable to fixed length algorithm for binary memoryless sources which converges to the entropy of the source at the optimal rate is known we study the problem of universal variable to fixed length coding for the class of markov sources with finite alphabets we give an upper bound on the performance of the code for large dictionary sizes and show that the code is optimal in the sense that no codes exist that have better asymptotic performance the optimal redundancy is shown to be h log log m log m where h is the entropy rate of the source and m is the code size this result is analogous to rissanen s 1984 result for fixed to variable length codes we investigate the performance of a variable to fixed coding method which does not need to store the dictionaries either at the coder or the decoder we also consider the performance of both these source codes on individual sequences for individual sequences we bound the performance in terms of the best code length achievable by a class of coders all the codes that we consider are prefix free and complete

fuzzy subrings and ideals redefined abstract v q fuzzy subrings and ideals of a ring are defined the concepts of v q fuzzy semiprime prime semi primary primary and maximal ideals are introduced and characterization of such fuzzy ideals are obtained v q fuzzy cosets determined by v q fuzzy ideals and v q fuzzy radicals of v q fuzzy ideals are discussed

irreducibility in algebraic fuzzy systems abstract inclusion invariant binary operations on an algebraic closure set system l are extended to a fuzzy l system the notions of prime maximal fuzzy l subset of a nonepty set x having truth values in a complete browerian lattice l are introduced and are completely determined in terms of pairs p where p is a prime maximal element of l and is an irreducible element dual atom of l it is observed that the space of all prime fuzzy l subsets of x together with its hull kernel topology is homeomorphic to the product of the spaces of prime elements of l and of irreducible elements of l equipped with hull kernel topologies also distributivity and irreducible elements of the fuzzy l system on x are discussed

the lattice of fuzzy ideals of a ring the set of all fuzzy ideals of a ring r is partially ordered with respect to the natural ordering of fuzzy sets in r it has been shown that this is a lattice but not a sublattice of the lattice of all fuzzy sets in r this lattice of all fuzzy ideals of r is complete distributive and bounded and has neither atoms nor dual atoms

rings with chain conditions abstract the aim of this paper is to find out some new results about fuzzy ideals to characterise artinian rings and noetherian rings by fuzzy ideals and to give alternative proofs of some well known results of artinian and noetherian rings by fuzzy ideal theoretic techniques

large time behavior of solutions to a nonlinear integro differential system abstract asymptotic behavior of solutions as t to the nonlinear integro differential system associated with the penetration of a magnetic field into a substance is studied initial boundary value problems with two kinds of boundary data are considered the first with homogeneous conditions on whole boundary and the second with non homogeneous boundary data on one side of lateral boundary the rates of convergence are given too

fuzzy ideals and fuzzy prime ideals of a ring abstract besides obtaining certain ring theoretic analogues some peculiarities of the fuzzy setting are also discussed in particular a study of i the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals ii the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms and iii the ring of all the fuzzy cosets of a fuzzy prime fuzzy maximal ideal is carried out

on fuzzy semiprimary ideals in this paper semiprimary ideals are redefined in terms of fuzzy points and it is proved that this definition is equivalent to that of kumar 1991 several results regarding semiprimary and fuzzy semiprimary ideals are derived it is shown by examples that the characterization of semiprimary ideals by prime radicals does not hold for this definition of fuzzy semiprimary ideals

galerkin finite element method for one nonlinear integro differential model galerkin finite element method for the approximation of a nonlinear integro differential equation associated with the penetration of a magnetic field into a substance is studied first type initial boundary value problem is investigated the convergence of the finite element scheme is proved the rate of convergence is given too the decay of the numerical solution is compared with the analytical results

a metatheorem for deriving fuzzy theorems from crisp versions a metatheorem is developed that allows theorems containing references to crisp substructures to be reformulated to refer to fuzzy substructures theorems concerning fuzzy substructures in semigroups rings and lattices are derived in concise fashion from analogous theorems that concern crisp substructures the modularity of the lattice of fuzzy normal subgroups of a group is concisely derived from the modularity of its lattice of crisp normal subgroups the fundamental tool used is a function that represents the algebra of fuzzy subsets of a set as a subdirect product of copies of the algebra of crisp subsets

finite difference scheme for one system of nonlinear partial integro differential equations abstract system of maxwell equations is considered reduction to the integro differential form is given existence uniqueness and large time behavior of solutions of the initial boundary value problem for integro differential model with two component and one dimensional case are studied finite difference scheme is investigated wider class of nonlinearity is studied than one has been investigated before freefem realization code and results of numerical experiments are given

spectrum of prime fuzzy ideals abstract a topological space called a spectrum of prime fuzzy ideals and denoted by fspec r is obtained by introducing a topology on the set of prime fuzzy ideals of a ring r the prime fuzzy ideals used here introduced by the author elsewhere are not necessarily two valued the topology introduced is compact and the usual nonfuzzy prime spectrum spec r is a subspace of fspec r and dense in it situations under which fuzzy spectrums of different rings are homeomorphic are discussed in particular fspec r is proved to be homeomorphic to fspec r frad r where frad r is the fuzzy prime radical of r the correspondence associating a ring r to the topological space fspec r is shown to define a contravarient functor from the category of rings with unity into the category of compact topological spaces

fuzzy maximal radical and primary ideals of a ring we introduce the concepts of fuzzy maximal ideal the fuzzy radical of a fuzzy ideal and fuzzy primary ideal of a ring we show that a fuzzy left right ideal a of a ring r is a fuzzy maximal ideal if and only a 0 1 and a x r a x a 0 is a maximal left right ideal of r we also show that a fuzzy ideal a of a commutative ring r with unity is a fuzzy primary ideal of r if and only a 0 1 a is two valued and a is a primary ideal of r

on fuzzy subnear rings and ideals abstract the aim of this paper is to introduce the notion of a fuzzy subnear ring to study fuzzy ideals of a near ring and to give some properties of fuzzy prime ideals of a near ring

fuzzy ideals and congruences of lattices general theory of algebraic fuzzy systems developed earlier by the authors is applied to extend the stone s prime ideal theorem to fuzzy ideals of distributive lattices a correspondence between fuzzy ideals and fuzzy congruences in a distributive lattice is obtained and certain important properties of these are discussed

fuzzy k ideals of semirings the aim of this paper is to introduce the notions of fuzzy k ideals prime fuzzy k ideals and semiregularity of semirings and to characterize prime fuzzy k ideals semiregular semirings noetherian semirings and artinian semirings by fuzzy k ideals

on fuzzy ideals and fuzzy bi ideals in semigroups abstract in this paper we give some properties of fuzzy ideals and fuzzy bi ideals of semigroups and characterize semigroups that are left duo left simple and semilattices of subsemigroups in terms of fuzzy ideals and fuzzy bi ideals

on a class of volterra nonlinear equations of parabolic type in this paper we show the backward uniqueness in time of solution for a class of volterra nonlinear equations of parabolic type we also give reasonable physical interpretation for our conclusion

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 prime ideals of a ring abstract this paper characterizes all fuzzy prime ideals p of an arbitrary ring r we show that a nonconstant fuzzy ideal p of r is prime if and only if p0 x r p x p 0 is a prime ideal of r p is two valued and p 0 1 examples are given showing that p0 is a prime ideal is not sufficient for p to be a fuzzy prime ideal and that p0 may be a prime ideal of r with the image of p being of arbitrary finite cardinality or being countably infinite

fuzzy prime spectrum of a ring ii abstract various topological properties of the fuzzy prime spectrum of a ring and certain of its subspaces as well as its functorial nature are discussed

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

nil radical of fuzzy ideal abstract the concepts of nil radical of a fuzzy ideal and primary fuzzy ideal are introduced and some fundamental propositions proved in particular a structural theorem for a primary fuzzy ideal and the associated prime fuzzy ideal

isomorphism theorems and fuzzy k ideals of k semirings we define and study the fuzzy maximal k ideal of a k semiring also we characterize the quotient k semiring r a of a k semiring r by a fuzzy maximal k ideal a and we have some isomorphism theorems in commutative k semirings and further we have some properties of the quotient k semiring r a of all cosets of a fuzzy semiprimary semiprime primary prime k ideal a in a commutative k semiring r

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

fuzzy cosets and some fuzzy radicals abstract an attempt is made to formulate fuzzy versions of jacobson and prime radicals of a ring certain basic results pertaining to the notions of jacobson and prime radicals of a ring are extended to the fuzzy setting in particular the following is obtained the jacobson radical prime radical of the ring of fuzzy cosets of in r is zero where r is a commutative ring with identity and is the fuzzy jacobson radical fuzzy prime radical of r

some results on a galois connection between lattices of l fuzzy ideals in this paper first we present a cl semigroup structure on l fuzzy ideals and conclude that the set of all l fuzzy quasi ideals is a cl monoid with zero then we obtain a galois connection between two cl semigroups of l fuzzy ideals of homomorphic rings finally by using galois connections we give the concepts of contracted and extended l fuzzy ideals and some related results are given

fuzzy semiprime ideals in semigroups abstract in this paper we shall introduce the notion of fuzzy semiprimality in a semigroup which is an extension of semiprimality in it and characterize a semigroup that is a semilattice of simple semigroups in terms of fuzzy semiprimality

fuzzy nil radicals and fuzzy semiprimary ideals of rings abstract an attempt is made to define a fuzzy semiprimary ideal by using the idea of the fuzzy nil radical of a fuzzy ideal the level ideals of a such a fuzzy ideal are examined and the ring of fuzzy cosets of such a fuzzy ideal is studied the algebraic nature of such fuzzy ideals under homomorphisms is also ascertained

global existence of solutions of volterra integrodifferential equations of parabolic type abstract sufficient conditions for the global existence of a strong solution of the equation u t t x t 0 k t s u x s x x ds f t x are given the kernel k satisfies r k z i k z and is increasing with sup p

large time behavior of solutions and finite difference scheme to a nonlinear integro differential equation the large time behavior of solutions and finite difference approximations of the nonlinear integro differential equation associated with the penetration of a magnetic field into a substance are studied asymptotic properties of solutions for the initial boundary value problem with homogeneous dirichlet boundary conditions is considered the rates of convergence are given too the convergence of the semidiscrete and the finite difference schemes are also proved

spectrum of prime l fuzzy h ideals of a hemiring we redefine the concept of prime fuzzy h ideals of a hemiring so that the fuzzy h ideals are not necessarily 2 valued we also introduce the concept of semiprime fuzzy h ideals a topological space called the spectrum of prime fuzzy h ideals of a commutative hemiring with unity has been obtained this topological space is compact and preserves isomorphisms between hemirings the correspondence associating a hemiring with its spectrum of prime fuzzy h ideals is shown to define a contravariant functor from the category of commutative hemirings with unity into the category of compact topological spaces the spectrum of crisp prime h ideals of the hemiring is a subspace which is dense in the spectrum of prime fuzzy h ideals valuation lattices for all the fuzzy sets in the paper are assumed to be complete heyting algebras

certain fuzzy ideals of rings redefined abstract an attempt is made to redefine fuzzy semiprimary fuzzy primary fuzzy prime and fuzzy maximal ideals of rings these fuzzy ideals are characterized by their level ideals fuzzy analogues of various results concerning semiprimary primary prime and maximal ideals are obtained the concept of a fuzzy nil radical of a fuzzy ideal is also formulated and discussed

a characterization of l fuzzy prime ideals abstract in this paper the concepts of a l fuzzy prime and a l fuzzy completely prime ideal are given and some fundamental lemmas are proved in particular bijections among some of the l fuzzy prime ideals of two homomorphic rings are constructed also a characterization of a l fuzzy prime ideal of a ring is given

prime fuzzy ideals over noncommutative rings in this paper we introduce prime fuzzy ideals over a noncommutative ring this notion of primeness is equivalent to level cuts being crisp prime ideals it also generalizes the one provided by kumbhojkar and bapat in 1993 which lacks this equivalence in a noncommutative setting semiprime fuzzy ideals over a noncommutative ring are also defined and characterized as intersection of primes this allows us to introduce the fuzzy prime radical and contribute to establish the basis of a fuzzy noncommutative ring theory

local examination of fuzzy intersection equations abstract we extend the primary representation theory for a fuzzy ideal of a commutative ring with identity in order to apply primary representation theory to a local examination of fuzzy intersection equations we show that a system of fuzzy intersection equations can be examined locally to obtain the general solution to the crisp part of the system

fuzzy ideals of artinian rings abstract the object of this paper is to study fuzzy ideals over artinian rings among others we prove that a ring with unity is left right artinian if and only if every fuzzy left right ideal of r is finite valued this gives a characterization of artinian rings in terms of fuzzy ideals mukherjee and sen 1987 proved that if every fuzzy ideal of a ring with unity is finite valued then the ring is noetherian we give an example to show that the converse of this theorem is not true we also show that certain results of pan 1987 are not true

on prime and primary fuzzy ideals and their radicals abstract the merits and demerits of various definitions of prime and primary fuzzy ideals are discussed and new definitions are suggested it is shown that the new definitions are more satisfactory the nil radical is defined as against the prime radical of a fuzzy ideal some theorems regarding the nil radical are proved it is shown that the two radicals coincide when the grade membership lattice is totally ordered

uniform decay rates for nonlinear viscoelastic marguerre von karman equations abstract in this paper we consider a model of nonlinear viscoelastic shallow shell that is referred to as the full marguerre von karman under the presence of long time memory we show that the energy functional associated with the system decays exponentially to zero as time goes to infinity

fractionary fuzzy ideals and dedekind domains we show that a finite valued fractionary fuzzy ideal of r with invertible level ideals has a finite minimal generating set as a fuzzy r submodule we also characterize dedekind domains in terms of the factorization of fuzzy ideals as products of prime fuzzy ideals and also in terms of the invertibility of certain fractionary fuzzy ideals we also examine the factorization of fractionary fuzzy ideals

a note on l fuzzy primary and semiprime ideals abstract in this note we introduce a definition for l fuzzy primary ideal in term of l fuzzy points and prove that this definition is equivalent to one of malik and mordeson 1991 where l 0 1 moreover l fuzzy semiprime ideals are introduced and some equivalent conditions of an l fuzzy semiprime ideal are proved

on prime quasi prime weakly quasi prime fuzzy left ideals of semigroups in this paper we introduce and study the prime quasi prime and weakly quasi prime fuzzy left ideals of a semigroup s we give characterizations of the largest fuzzy ideal i f contained in a fuzzy left ideal f of a semigroup s and the largest fuzzy subsemigroup i f such that f is a fuzzy ideal of i f finally strongly semisimple semigroups are characterized by quasi prime fuzzy left ideals

fuzzy generalized bi ideals in semigroups abstract we introduce the concept of a fuzzy generalized bi ideal of a semigroup which is an extension of the concept of a fuzzy bi ideal and of a nonfuzzy bi ideal and a nonfuzzy ideal of a semigroup and characterize regular semigroups and both intraregular and left quasiregular semigroups in terms of fuzzy generalized bi ideals

r primary l representations of l ideals we examine the basic properties of prime l ideals primary l ideals and r radicals of l ideals where l is a completely distributive lattice we determine to what extent every l ideal has an r primary l representation