fuzzy control on the basis of equality relations with an example from idle speed control the way engineers use fuzzy control in real world applications is often not coherent with an understanding of the control rules as logical statements or implications in most cases fuzzy control can be seen as an interpolation of a partially specified control function in a vague environment which reflects the indistinguishability of measurements or control values in this paper the authors show that equality relations turn out to be the natural way to represent such vague environments and they develop suitable interpolation methods to obtain a control function as a special case of our approach the authors obtain mamdani s model and can justify the inference mechanism in this model and the use of triangular membership functions not only for the reason of simplified computations and they can explain why typical fuzzy partitions are preferred the authors also obtain a criterion for reasonable defuzzification strategies the fuzzy control methodology introduced in this paper has been applied successfully in a case study of engine idle speed control for the volkswagen golf gti

fuzzy equivalence relations and their equivalence classes in this paper we investigate various properties of equivalence classes of fuzzy equivalence relations over a complete residuated lattice we give certain characterizations of fuzzy semi partitions and fuzzy partitions over a complete residuated lattice as well as over a linearly ordered complete heyting algebra in the latter case for a fuzzy equivalence relation over a linearly ordered complete heyting algebra we construct an algorithm for calculation of a minimal family of its equivalence classes which generates it most of the presented results are new but some of them are generalizations of known results given in a way which simplifies and clarifies them

brzozowski type determinization for fuzzy automata abstract in this paper we adapt the well known brzozowski determinization method to fuzzy automata this method gives better results than all previously known methods for determinization of fuzzy automata developed by b lohlavek 4 li and pedrycz 20 ignjatovic et al 15 and jancic et al 18 namely as in the case of ordinary nondeterministic automata brzozowski type determinization of a fuzzy automaton results in a minimal crisp deterministic fuzzy automaton equivalent to the starting fuzzy automaton and we show that there are cases when all previous methods result in infinite automata while brzozowski type determinization results in a finite one the paper deals with fuzzy automata over complete residuated lattices but identical results can also be obtained in a more general context for fuzzy automata over lattice ordered monoids and even for weighted automata over commutative semirings

fuzzy homomorphisms of rings abstract in this paper we introduce the concept of fuzzy homomorphisms of rings we prove for fuzzy homomorphisms analogues of the homomorphism theorems of ordinary ring homomorphisms

further improvements of determinization methods for fuzzy finite automata in this paper we provide further improvements of determinization methods for fuzzy finite automata these methods perform better than all previous determinization methods for fuzzy finite automata developed by b lohlavek 3 li and pedrycz 38 ignjatovic et al 25 and jancic et al 33 in the sense that they produce smaller automata while require the same computation time the only exception is the brzozowski type determinization algorithm developed recently by jancic and iric 34 which produces a minimal crisp deterministic fuzzy automaton but the algorithms created here can also be used within the brzozowski type algorithm and improve its performance

fuzzy homomorphism and fuzzy isomorphism abstract in this paper we introduce the concepts of fuzzy homomorphism and fuzzy isomorphism between two fuzzy groups by a natural way and study some of their properties we prove the decomposition theorems for fuzzy homomorphism and fuzzy isomorphism lastly using the concept of fuzzy quotient group we give the fuzzy homomorphism isomorphism theorems of fuzzy groups

fuzzy homomorphisms of algebras in this paper we consider fuzzy relations compatible with algebraic operations which are called fuzzy relational morphisms in particular we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations called uniform fuzzy relational morphisms and those which are partially uniform f functions called fuzzy homomorphisms both uniform fuzzy relations and partially uniform f functions were introduced in a recent paper by us uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras we give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms we establish certain relationships between them and fuzzy congruences and we prove homomorphism and isomorphism theorems concerning them we also point to some applications of uniform fuzzy relational morphisms

uniform fuzzy relations and fuzzy functions in this paper we introduce and study the concepts of a uniform fuzzy relation and a partially uniform f function we give various characterizations and constructions of uniform fuzzy relations and uniform f functions we show that the usual composition of fuzzy relations is not convenient for f functions so we introduce another kind of composition and we establish a mutual correspondence between uniform f functions and fuzzy equivalences we also give some applications of uniform fuzzy relations in approximate reasoning especially in fuzzy control and we show that uniform fuzzy relations are closely related to the defuzzification problem

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

limit hamilton jacobi isaacs equations for singularly perturbed zero sum differential games abstract a singularly perturbed zero sum differential game with full information is considered upper and lower value functions of this game are shown to have limits as the singular perturbations parameter tends to zero these limits are established to coincide with viscosity solutions of some hamilton jacobi type equations a special case and two examples are considered to illustrate the general results

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

weak bisimulations for fuzzy automata abstract forward and backward bisimulations have been introduced recently by ciric et al 2012 9 as a means for modeling the equivalence between states of fuzzy automata and approximating the language equivalence as well as for use in the state reduction of fuzzy automata the main aim of the present paper is to introduce two new kinds of bisimulations weak forward and backward bisimulations which provide better approximations of the language equivalence than forward and backward bisimulations and when employed in the state reduction they provide better reductions we give procedures for deciding whether there exist weak forward and backward simulations and bisimulations and for computing the greatest ones whenever they exist using weak bisimulations in conjunction with the concept of a uniform fuzzy relation we determine necessary and sufficient conditions under which two fuzzy automata are weak bisimulation equivalent we also characterize uniform weak backward and forward bisimulations between two fuzzy automata in terms of isomorphisms between their nerode and reverse nerode automata

asymptotic properties of input output operators norm associated with singularly perturbed systems with multiplicative white noise in this paper we study the problem of the asymptotic property of the norm of input output operators related to a class of singularly perturbed stochastic linear systems the system is under perturbation of multiplicative white noise by using reduction order and boundary layer techniques it is shown that the norm of the operator of the perturbed system is less than a given number gamma when the small perturbation varepsilon tends to zero if both the related norms of the reduced subsystem and the boundary layer subsystem are less than gamma furthermore a stabilizing robust controller is designed which is independent of perturbation varepsilon

fuzzy quotient algebras and fuzzy factor congruences fuzzy quotient algebras are defined some homomorphism theorems are given fuzzy factor congruences and fuzzy factorable algebras are defined and some basic theorems are proved

on the conservativeness and the recurrence of symmetric jump diffusions abstract sufficient conditions for a symmetric jump diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process a number of examples are presented to illustrate the optimality of these conditions in particular the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components

singularly perturbed ordinary differential equations with dynamic limits coupled slow and fast motions generated by ordinary differential equations are examined the qualitative limit behaviour of the trajectories as the small parameter tends to zero is sought after invariant measures of the parametrised fast flow are employed to describe the limit behaviour rather than algebraic equations which are used in the standard reduced order approach in the case of a unique invariant measure for each parameter the limit of the slow motion is governed by a chattering type equation without the uniqueness the limit of the slow motion solves a differential inclusion the fast flow in turn converges in a statistical sense to the direct integral respectively the set valued direct integral of the invariant measures

fuzzy homomorphism and f fuzzy subgroups generated by a fuzzy subset abstract the main objects of this paper are 1 i to characterise t level image f t h of a subgroup h g 1 under f and t level inverse image f t 1 h of a subgroup s g 2 and 2 ii to study the t level f fuzzy subgroup generated by a fuzzy subset g 1 v of g 1 and a fuzzy subset g 2 v of g 2 where f g 1 g 2 is a fuzzy group homomorphism

nondeterministic automata equivalence bisimulations and uniform relations in this paper we study the equivalence of nondeterministic automata pairing the concept of a bisimulation with the recently introduced concept of a uniform relation in this symbiosis uniform relations serve as equivalence relations which relate states of two possibly different nondeterministic automata and bisimulations ensure compatibility with the transitions initial and terminal states of these automata we define six types of bisimulations but due to the duality we discuss three of them forward backward forward and weak forward bisimulations for each of these three types of bisimulations we provide a procedure which decides whether there is a bisimulation of this type between two automata and when it exists the same procedure computes the greatest one we also show that there is a uniform forward bisimulation between two automata if and only if the factor automata with respect to the greatest forward bisimulation equivalences on these automata are isomorphic we prove a similar theorem for weak forward bisimulations using the concept of a weak forward isomorphism instead of an isomorphism we also give examples that explain the relationships between the considered types of bisimulations

fuzzy homomorphism and algebraic structures abstract the object of this paper is to prove an analogue of the fundamental theorem of homomorphism and the second isomorphism theorem for fuzzy homomorphisms we also study the fuzzy subgroup generated by a fuzzy subset

determinization of fuzzy automata with membership values in complete residuated lattices in this paper we introduce a new method for determinization of fuzzy finite automata with membership values in complete residuated lattices in comparison with the previous methods developed by belohlavek r belohlavek determinism and fuzzy automata information sciences 143 2002 205 209 and li and pedrycz y m li w pedrycz fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids fuzzy sets and systems 156 2005 68 92 our method always gives a smaller automaton and in some cases when the previous methods result in infinite automata our method can result in a finite one we also show that determinization of fuzzy automata is closely related to fuzzy right congruences on a free monoid and fuzzy automata associated with them and in particular to the concept of the nerode s fuzzy right congruence of a fuzzy automaton which we introduce and study here

h sup spl infin optimal control for singularly perturbed systems ii imperfect state measurements for pt i see ibid vol 29 no 2 p 401 423 1993 in this paper we study the h sup spl infin optimal control of singularly perturbed linear systems under general imperfect measurements for both finite and infinite horizon formulations using a differential game theoretic approach we first show that as the singular perturbation parameter say spl epsiv 0 approaches zero the optimal disturbance attenuation level for the full order system under a quadratic performance index converges to a value that is bounded above by and in some cases equal to the maximum of the optimal disturbance attenuation levels for the slow and fast subsystems under appropriate slow and fast quadratic cost functions with the bound being computable independently of e and knowing only the slow and fast dynamics of the system we then construct a controller based on the slow subsystem only and obtain conditions under which it delivers a desired performance level even though the fast dynamics are completely neglected the ultimate performance level achieved by this slow controller can be uniformly improved upon however by a composite controller that uses some feedback from the output of the fast subsystem we construct one such controller via a two step sequential procedure which uses static feedback from the fast output and dynamic feedback from an appropriate slow output each one obtained by solving appropriate spl epsiv independent lower dimensional h sup spl infin optimal control problems under some informational constraints we provide a detailed analysis of the performance achieved by this lower dimensional spl epsiv independent composite controller when applied to the full order system and illustrate the theory with some numerical results on some prototype systems

fuzzy group based on fuzzy binary operation abstract a new kind of fuzzy group based on fuzzy binary operations is proposed the concepts of fuzzy subgroup normal fuzzy subgroup factor fuzzy group and fuzzy group homomorphisms are introduced finally the fundamental homomorphism theorem of fuzzy group is obtained

quotients with respect to similarity relations abstract the problem of the quotient w r t a similarity relation is closely related to poincare s paradox and central problems in cluster analysis amazingly this problem remains open for more than 30 years as a generalization of strong m categories we introduce the concept of m valued sets which leads to the definition of the category m set in m set we give a complete solution of the quotient problem

limit hamilton jacobi isaacs equations for singularly perturbed zero sum dynamic discrete time games in this paper we study a singularly perturbed zero sum dynamic game with full information we introduce the upper lower value function of the dynamic game in which the minimizer maximizer can be guaranteed if at the beginning of each interval his move the choice of decision precedes the move of the maximizer minimizer we show that when the singular perturbations parameter tends to zero the upper lower value function of the dynamic game has a limit which coincides with a viscosity solution of a hamilton jacobi isaacs type equation two examples are given to demonstrate the potential of the proposed technique

construction of fuzzy automata from fuzzy regular expressions li and pedrycz have proved fundamental results that provide different equivalent ways to represent fuzzy languages with membership values in a lattice ordered monoid and generalize the well known results of the classical theory of formal languages in particular they have shown that a fuzzy language over an integral lattice ordered monoid can be represented by a fuzzy regular expression if and only if it can be recognized by a fuzzy finite automaton however they did not give any efficient method for constructing an equivalent fuzzy finite automaton from a given fuzzy regular expression in this paper we provide such an efficient method transforming scalars appearing in a fuzzy regular expression ainto letters of the new extended alphabet we convert the fuzzy regular expression ato an ordinary regular expression a r then starting from an arbitrary nondeterministic finite automaton a that recognizes the language a r represented by the regular expression a r we construct fuzzy finite automata a a and a a r with the same or even less number of states than the automaton a which recognize the fuzzy language a represented by the fuzzy regular expression a the starting nondeterministic finite automaton a can be obtained from a r using any of the well known constructions for converting regular expressions to nondeterministic finite automata such as glushkov mcnaughton yamada s position automaton brzozowski s derivative automaton antimirov s partial derivative automaton or ilie yu s follow automaton

fuzzy congruences on inverse semigroups abstract we introduce the notion of a quotient semigroup s induced by a fuzzy congruence relation on a semigroup s and give homomorphism theorems with respect to the fuzzy congruence we also introduce the notion of an idempotent separating fuzzy congruence a group fuzzy congruence on inverse semigroups and give some of their properties on inverse semigroups

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

conservativeness criteria for generalized dirichlet forms abstract we develop sufficient analytic conditions for conservativeness of non sectorial perturbations of symmetric dirichlet forms which can be represented through a carre du champ on a locally compact separable metric space these form an important subclass of generalized dirichlet forms which were introduced in 21 in case there exists an associated strong feller process the analytic conditions imply conservativeness i e non explosion of the associated process in the classical probabilistic sense as an application of our general results on locally compact separable metric state spaces we consider a generalized dirichlet form given on a closed or open subset of r d which is given as a divergence free first order perturbation of a symmetric energy form then using volume growth conditions of the carre du champ and the non sectorial first order part we derive an explicit criterion for conservativeness we present several concrete examples which relate our results to previous ones obtained by different authors in particular we show that conservativeness can hold for a large variance if the anti symmetric part of the drift is strong enough to compensate it this work continues our previous work on transience and recurrence of generalized dirichlet forms

fuzzy finite automata and fuzzy regular expressions with membership values in lattice ordered monoids we study fuzzy finite automata in which all fuzzy sets are defined by membership functions whose codomain forms a lattice ordered monoid l for these l fuzzy finite automata l ffa for short we provide necessary and sufficient conditions for the extendability of the state transition function it is shown that nondeterministic l ffa nl ffa for short are more powerful than deterministic l ffa dl ffa for short then we give necessary and sufficient conditions for the simulation of an nl ffa by an equivalent dl ffa next we turn to the closure properties of languages defined by l ffas we establish closure under the regular operations and provide conditions for closure under intersection and reversal in particular we show that the family of fuzzy languages accepted by dl ffas is not closed under kleene closure operation and the family of fuzzy languages accepted by nl ffas is not closed under complement operation furthermore we introduce the notions of l fuzzy regular expressions and give the kleene theorem for nl ffas the description of dl ffas by l fuzzy regular expressions is also given finally we investigate the level structures of l ffas our results provide some insight as to what extend properties of l ffas and their languages depend on the algebraic properties of l

geometry and analysis of dirichlet forms ii abstract given a regular strongly local dirichlet form e under assumption that the lower bound of the ricci curvature of bakry emery the local doubling and local poincare inequalities are satisfied we obtain that i the intrinsic differential and distance structures of e coincide ii the cheeger energy functional ch d e is a quadratic norm this shows that ii is necessary for the riemannian ricci curvature defined by ambrosio gigli savare to be bounded from below this together with some recent results of ambrosio gigli savare yields that the heat flow gives a gradient flow of boltzman shannon entropy under the above assumptions we also obtain an improvement on kuwada s duality theorem for dirichlet forms under the assumptions of doubling and poincare inequalities finally dirichlet forms are constructed to show that doubling and poincare inequalities are not enough to obtain either i or ii above that is the lower bound of the bakry emery curvature condition is essential

algebras with fuzzy equalities an algebra with fuzzy equality is a set with operations on it that is equipped with similarity i e a fuzzy equivalence relation such that each operation f is compatible with described verbally compatibility says that each f yields similar results if applied to pairwise similar arguments on the one hand algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic and have been studied from this point of view before on the other hand they are the formal counterpart to the intuitive idea of having functions that are not allowed to map similar objects to dissimilar ones the present paper aims at developing fundamental points of algebras with fuzzy equalities we introduce the notion of an algebra with fuzzy equality present natural examples compare the notion with other approaches and introduce and develop basic structural notions subalgebras morphisms products direct unions in a follow up paper vychodil direct limits and reduced products of algebras with fuzzy equalities submitted for publication we deal with advanced topics in algebras with fuzzy equalities direct limits reduced products

a behavioral distance for fuzzy transition systems in contrast with the existing approaches to exact bisimulation for fuzzy systems we introduce a robust notion of behavioral distance to measure the behavioral similarity of nondeterministic fuzzy transition systems which are a generalization of fuzzy automata this behavioral distance provides a quantitative analogue of bisimilarity and is defined as the greatest fixed point of a suitable monotonic function the behavioral distance has the important property that two systems are at zero distance if and only if they are bisimilar moreover for any given threshold we find that systems with behavioral distances bounded by the threshold are equivalent in addition we show that two system combinators parallel composition and product are nonexpansive with respect to our behavioral distance which makes compositional verification possible the theory developed here is applicable to the quantitative verification approximate reduction and reliability analysis of fuzzy transition systems

reduction of fuzzy automata by means of fuzzy quasi orders abstract in our recent paper we have established close relationships between state reduction of a fuzzy automaton and resolution of a particular system of fuzzy relation equations in that paper we have also studied reductions by means of those solutions which are fuzzy equivalences in this paper we will see that in some cases better reductions can be obtained using the solutions of this system that are fuzzy quasi orders generally fuzzy quasi orders and fuzzy equivalences are equally good in the state reduction but we show that right and left invariant fuzzy quasi orders give better reductions than right and left invariant fuzzy equivalences we also show that alternate reductions by means of fuzzy quasi orders give better results than alternate reductions by means of fuzzy equivalences furthermore we study a more general type of fuzzy quasi orders weakly right and left invariant ones and we show that they are closely related to determinization of fuzzy automatons we also demonstrate some applications of weakly left invariant fuzzy quasi orders in conflict analysis of fuzzy discrete event systems

suboptimization of singularly perturbed control systems a technique different from the boundary layer method is developed to deal with singularly perturbed optimal control problems the technique is applicable in particular in the case when the optima

bisimulations for fuzzy automata bisimulations have been widely used in many areas of computer science to model equivalence between various systems and to reduce the number of states of these systems whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata we prove that a uniform fuzzy relation between fuzzy automata a and b is a forward bisimulation if and only if its kernel and co kernel are forward bisimulation fuzzy equivalence relations on a and b and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations as a consequence we get that fuzzy automata a and b are ufb equivalent i e there is a uniform forward bisimulation between them if and only if there is a special isomorphism between the factor fuzzy automata of a and b with respect to their greatest forward bisimulation fuzzy equivalence relations this result reduces the problem of testing ufb equivalence to the problem of testing isomorphism of fuzzy automata which is closely related to the well known graph isomorphism problem we prove some similar results for backward forward bisimulations and we point to fundamental differences because of the duality with the studied concepts backward and forward backward bisimulations are not considered separately finally we give a comprehensive overview of various concepts on deterministic nondeterministic fuzzy and weighted automata which are related to bisimulations

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations in this paper we study systems of fuzzy relation inequalities and equations of the form u v i v i u i i where u is an unknown and v i i i are given fuzzy relations the dual systems v i u u v i i i their conjunctions the systems of the form u v i v i u i i and certain special types of these systems we call them weakly linear systems for each weakly linear system with a complete residuated lattice as the underlying structure of truth values we prove the existence of the greatest solution and we provide an algorithm for computing the greatest solution which works whenever the underlying complete residuated lattice is locally finite otherwise we determine some sufficient conditions under which the algorithm works the algorithm is iterative and each its single step can be viewed as solving of a particular linear system weakly linear systems emerged from the fuzzy automata theory but we show that they also have important applications in other fields e g in the concurrency theory and social network analysis

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

computation of the greatest simulations and bisimulations between fuzzy automata recently two types of simulations forward and backward simulations and four types of bisimulations forward backward forward backward and backward forward bisimulations between fuzzy automata have been introduced if there is at least one simulation bisimulation of some of these types between the given fuzzy automata it has been proved that there is the greatest simulation bisimulation of this kind in the present paper for any of the above mentioned types of simulations bisimulations we provide an efficient algorithm for deciding whether there is a simulation bisimulation of this type between the given fuzzy automata and for computing the greatest one whenever it exists the algorithms are based on the method developed in ignjatovic et al on the greatest solutions to weakly linear systems of fuzzy relation inequalities and equations fuzzy sets syst 161 2010 3081 3113 which comes down to the computing of the greatest post fixed point contained in a given fuzzy relation of an isotone function on the lattice of fuzzy relations

lattice of fuzzy subalgebras and closure systems in i x abstract let x be a non empty set and i be the unit interval of real numbers the elements of i x are called fuzzy subsets of x i x is a lattice with the pointwise ordering of i now if a x f is an universal algebra with a specified set f of finitary operations on x then each operation can be extended to a finitary operation on i x an element of i x is defined to be a fuzzy subalgebra of a using these extended operations we also define algebraic closure system in i x we then show that the lattice of fuzzy subalgebras of a is an algebraic closure system in i x

a note on fuzzy subgroups and fuzzy homomorphism on introduit la notion d homomorphisme flou entre deux groupes et on etudie son effet sur des sous groupes flous

determinism and fuzzy automata the concept of a fuzzy automaton as studied in the literature is a direct generalization of the classical concept of a non deterministic automaton in this paper we consider the question of whether there is a deterministic counterpart to the notion of a fuzzy automaton we propose a definition of deterministic fuzzy automata and show that they are equally powerful as fuzzy automata compared to the classical notion of a deterministic automaton the deterministic fuzzy automaton defined in our paper differs essentially only in that the final states form a fuzzy set moreover we show that deterministic fuzzy automata are equally powerful as nested systems of deterministic automata

a theory of vague lattices based on many valued equivalence relations i general representation results the present work introduces a new and general theory of ordering relations and lattices based on many valued equivalence relations under the name vague ordering relations and vague lattices respectively representations and constructions of vague ordering relations and vague lattices are the main subjects of this paper and various desirable results in this direction are established furthermore algebraic characterizations of vague lattices are studied

fuzzy congruence relations abstract if a x f is an universal algebra and if r x denotes the set of all fuzzy subsets of x x i e fuzzy relations on x then the algebraic operations fg f on x are extended to algebraic operations f on r x f is an algebra similar to a it is called relational algebra a fuzzy equivalence relation which is at the same time a fuzzy subalgebra of r x f is called a fuzzy congruence relation on x in this paper we study fuzzy congruence relations amongst other things we prove that the set of all fuzzy congruence relations on an algebra is a complete lattice and an algebraic closure system