the reduction of binary fuzzy relations and its applications this paper introduces the concept of t reduction of binary fuzzy relations and establishes its properties it presents an approach to providing the one valued restoration of any boundary transitive and t asymmetric binary fuzzy relation by its t reduction a theoretical basis for this approach is established showing that t reduction is the smallest fuzzy relation whose t transitive closure coincides with the given binary fuzzy relation and where an nxn fuzzy relation on a finite universe with cardinality n can be represented by n 1 values this approach opens up an original way for constructing the membership functions of certain binary fuzzy relations important examples for the application of this concept are given

partial ordering in l underdeterminate sets abstract the purpose of this paper is to improve results on fuzzy partial orderings obtained by zadeh in 9 we overcome the difficulties connected with the axioms of antisymmetry and linearity moreover if the underlying lattice l is a complete residuated lattice we establish a szpilrajn theorem i e any l fuzzy partial ordering has a linear extension in opposition to zadeh s our point of view is that an axiom of antisymmetry without a reference to a concept of equality is meaningless therefore we first introduce the category lus cf 2 which can be considered as a mathematical model of fuzzy equality and subsequently we specify the axioms of l fuzzy partial orderings with respect to the frame given by lus the axioms we use clearly display the usefulness of having a zadeh like complementation and as a consequence the usefulness of a positivistic and nonintuitionistic frame of study an example concerning l rn which we give clearly shows that the lus version of the szpilrajn theorem cannot be reduced to a fuzzification of an already existing theorem but provides us with additional information

influence of errors on the two microphone method for measuring acoustic properties in ducts using the two microphone method acoustic properties in ducts as for example reflection coefficient and acoustic impedance can be calculated from a transfer function measurement between two microphones in this paper a systematic investigation of the various measurement errors that can occur and their effect on the calculated quantities is made the input data for the calculations are the measured transfer function the microphone separation and the distance between one microphone and the sample first errors in the estimate of the transfer function are treated conclusions concerning the most favorable measurement configuration to avoid these errors are drawn next the length measurement errors are treated measurements were made to study the question of microphone interference the influence of errors on the calculated quantities has been investigated by numerical simulation from this conclusions are drawn on the useful frequency range for a given microphone separation and on the magnitude of errors to expect for different cases

the cutting of compositions this paper investigates the cutability of the fuzzy relational compositions the sup t composition the triangular compositions of bandler and kohout and their improved versions the theoretical and practical importance of fuzzy cuts is illustrated by means of a careful selection of results from the fuzzy literature some new potential properties of fuzzy implications operators and triangular norms that play an important role in this cutability study are introduced and discussed extensively the paper concludes with a study of the fuzzy cutability of the sup t composition

transitivity bounds in additive fuzzy preference structures transitivity plays a crucial role in preference modeling and related fields in this paper we discuss this property in the general context of additive fuzzy preference structures of particular interest is the decomposition of a large preference relation r in its symmetric part i indifference relation and its asymmetric part p strict preference relation by means of a so called indifference generator i given the type of transitivity of a large preference relation r w r t a conjunctor and a generator we establish basic lower bounds and general upper bounds on the transitivity of p and i these bounds are due to the careful design of generic counterexamples moreover we identify the situations in which these bounds are effectively reached thereby establishing connections with interesting properties such as dominance bisymmetry the 1 lipschitz property and rotation invariance

fuzzy sets and vague environments abstract in this paper we propose a natural approach to handle imprecise numbers as they arise for example from measurements fuzzy sets turn out to be a canonical representation for such imprecise numbers that are induced by taking different tolerance or error bounds into account fuzzy sets are induced by scaling factors that describe the magnitude of the imprecision on the other the scaling factors can be derived from given fuzzy sets so that we have a correspondence between scaling factors and fuzzy sets when these concepts are applied to control problems the max min rule is rediscovered as an interpolation technique viewing fuzzy control as an interpolation technique in vague environments enables us to validate various concepts for the design and tuning of fuzzy controllers and also suggests new methods based on clear semantics

t partitions of the real line generated by idempotent shapes abstract the idea of generating fuzzy numbers as equivalence classes of particular t equivalences on the real line r is fully exploited scales or generators are used to define certain pseudo metrics on r by means of a shape function these pseudo metrics are then transformed into binary fuzzy relations on r shapes leading to t equivalences and hence to a class of fuzzy numbers forming a t partition of r are completely characterized in the case of a continuous generator this characterization problem is shown to be closely related to determining the idempotents w r t the t addition of fuzzy numbers

statistical inference method for liner impedance eduction with a shear grazing flow this paper presents a statistical inference method for impedance eduction in a flow duct facility the acoustic impedance is recast into a random variable and bayes s theorem is used to obtain the

on the existence and construction of t transitive closures in this paper it is shown that any fuzzy relation r on an arbitrary universe x has a t transitive closure the triangular norm t involved is not subject to any conditions this existential result can be turned into an explicit expression in the case of a left continuous triangular norm t furthermore this continuity restriction vanishes in the case of a finite universe x in that case the explicit expression can also be operationalized

on the compositional characterization of complete fuzzy pre orders complete pre orders can be characterized in terms of the transitivity of the corresponding strict preference and indifference relations in this paper we investigate this characterization in a fuzzy setting we consider two types of completeness weak completeness and strong completeness and decompose a fuzzy pre order by means of an indifference generator in particular a frank t norm in the weakly complete case we identify the strongest type of transitivity of the indifference and strict preference relations in function of the generator used for constructing them in the strongly complete case we lay bare a stronger type of transitivity of the strict preference relation we conclude the paper with a rather negative result there is no hope to obtain a compositional characterization of weakly complete fuzzy pre orders and hence also not of fuzzy pre orders in general

a compendium of fuzzy weak orders representations and constructions the present paper gives a state of the art overview of representation and construction results for fuzzy weak orders we do not assume that the underlying domain is finite instead we concentrate on results that hold in the most general case that the underlying domain is possibly infinite this paper presents three fundamental representation results each of which also provides a construction method i score function based representations ii inclusion based representations iii representations by decomposition into crisp linear orders and fuzzy equivalence relations which also facilitates a pseudo metric based construction

linear differential systems with conditionally integrable coefficients abstract for perturbations conditionally integrable levinson s theorems are obtained and applied to almost constant systems

a top down algorithm for generating the hasse tree of a fuzzy preorder closure this paper describes a new top down algorithm for the stepwise generation of the different levels or hasse diagrams of the hasse tree associated to the fuzzy preorder closure min transitive closure of a given reflexive binary fuzzy relation the algorithm is based upon a recently established weight driven method for computing the min transitive closure of a reflexive binary fuzzy relation the way in which this method gradually establishes the fuzzy preorder closure implies that for the generation of a specific level of the hasse tree the newly proposed algorithm does not require the complete computation of this closure

fast solving method of fuzzy relational equation and its application to lossy image compression reconstruction a fast solving method of the solution for max continuous t norm composite fuzzy relational equation of the type g i j r sup t spl square a sub i sup t spl square b sub j i 1 2 i j 1 2 j where a sub i spl isin f x x x sub 1 x sub 2 x sub m bj spl isin f y y y sub 1 y sub 2 y sub n r spl isin f x spl times y and spl square max continuous t norm composition is proposed it decreases the computation time ijmn l t p to jm i n l p where l t and p denote the computation time of min t norm and relative pseudocomplement operations respectively by simplifying the conventional reconstruction equation based on the properties of t norm and relative pseudocomplement the method is applied to a lossy image compression and reconstruction problem where it is confirmed that the computation time of the reconstructed image is decreased to 1 335 6 the compression rate being 0 0351 and it achieves almost equivalent performance for the conventional lossy image compression methods based on discrete cosine transform and vector quantization

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

on identification in fuzzy systems and its applications in control problems abstract the problem of identification in fuzzy systems described by the use of fuzzy equation is considered the identification method and its performance index is also presented the formal procedure of the identification algorithm is illustrated by means of a numerical example the possibility of using the proposed algorithm for the solution of a control problem is given as well

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

maxima and minima in fuzzified linear orderings the paper studies the notions of fuzzy maxima minima suprema and infima in the setting of fuzzified linear orderings the main result shows that in fuzzified linear orders satisfying certain reasonable assumptions the fuzzy maxima and minima of all non empty fuzzy sets are non empty

a transitivity analysis of bipartite rankings in pairwise multi class classification many multi class classification algorithms in statistics and machine learning typically combine several binary classifiers in order to construct an overall classifier in the popular pairwise ensemble one classifier is built for each pair of classes resulting in pairwise bipartite rankings in contrast ordinal regression algorithms consider a single ranking function for several ordered classes it is known in the literature that pairwise ensembles can be useful for ordinal regression however can single ranking models make a contribution to multi class classification the answer to this question should be affirmative as supported by theoretical results presented in this article we conduct a formal analysis of the consistency of pairwise bipartite rankings by uncovering the conditions under which they can be equivalently expressed in terms of a single ranking similar to the utility representability of pairwise preference relations it turns out that transitivity plays a crucial role in the characterization of the ranking representability of pairwise bipartite rankings to this end we introduce the new concepts of strict ranking representability a restrictive condition that can be verified easily and auc ranking representability a practically more useful condition that is more difficult to verify however the link between pairwise bipartite rankings and dice games allows us to formulate necessary transitivity conditions for auc ranking representability a sufficient condition on the other hand is obtained by introducing a new type of transitivity that can be verified by solving an integer quadratic program

representations and constructions of similarity based fuzzy orderings this contribution advocates the dissemination of similarity based fuzzy orderings a generalized concept of fuzzy orderings which appears to be unavoidable as soon as the classical correspondences between preorderings equivalence relations and orderings are taken into account in contrast to earlier papers on the topic this paper is focused on representation and construction results which once more demonstrate the feasibility of the proposed concept

additive decomposition of fuzzy pre orders fuzzy pre orders reflexive and min transitive fuzzy relations constitute an important class of fuzzy relations by means of an indifference generator a fuzzy pre order can be decomposed additively into two parts an indifference relation and a strict preference relation when using a frank t norm as indifference generator we fully characterize the transitivity of these parts only in case the minimum operator is used as generator both parts are min transitive the transitivity of the indifference relation is determined by the frank t norms while the transitivity of the strict preference relation is determined by transforms of the nilpotent minimum t norm

discontinuous galerkin method for the computation of acoustic modes in lined flow ducts with rigid splices a discontinuous galerkin method for the discretisation of the linearised euler equations is derived for the computation of acoustic modes in an infinite duct of arbitrary geometric shape in presence of axial sheared flow validations are performed with analytical solutions in the case of a uniformly lined duct and comparisons are done with the literature in the case of a lined cylindrical duct with rigid splices we take advantage of the discontinuous formulation to study the behaviour of the acoustic eigenfunctions in the vicinity of the liner splice transition where singular behaviours of the acoustic perturbation are found

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures ii the identity case abstract it has been shown in the first part of this paper that the concept of a fuzzy preference structure is only meaningful provided that the de morgan triplet involved contains a continuous archimedean triangular norm having zero divisors or hence a transform of the ukasiewicz triangular norm in this second part additional arguments for this statement are supplied in what we call the identity case is the identity mapping and the involutive negator is the standard negator first it is shown that the use of a continuous non archimedean triangular norm having zero divisors in the definition of a fuzzy preference structure indeed is possible secondly using such a triangular norm implies that at least in the square 0 4 5 2 it actually behaves as the ukasiewicz triangular norm furthermore an important transformation theorem indicates that any fuzzy preference structure with respect to a de morgan triplet containing such a continuous non archimedean triangular norm having zero divisors can be transformed into a fuzzy preference structure with respect to the standard ukasiewicz triplet these additional arguments conclude in a convincing way our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

fuzzy preference structures without incomparability abstract in this paper we establish important relationships between the basic properties of the components of a fuzzy preference structure without incomparability this study is carried out for the fuzzy preference structures introduced recently by de baets van de walle and kerre a set of remarkable theorems gives detailed insight in the relationships between the sup t transitivity of the fuzzy preference and indifference relations and the sup t transitivity of the fuzzy large preference relation several paths of thought involving t norms with or without zero divisors are explored and where required illustrative counterexamples confirm the falsity of certain implications finally we introduce the t n ferrers property of a binary fuzzy relation and show that the fuzzy preference and fuzzy large preference relations share certain types of this ferrers property

a dynamical systems result on asymptotic integration of linear differential systems abstract we are interested in the asymptotic integration of linear differential systems of the form x t r t x where is diagonal and r l p t 0 for p 1 2 our dichotomy condition is in terms of the spectrum of the omega limit set our results include examples that are not covered by the hartman wintner theorem

a formal study of linearity axioms for fuzzy orderings abstract this contribution is concerned with a detailed investigation of linearity axioms for fuzzy orderings different existing concepts are evaluated with respect to three fundamental correspondences from the classical case linearizability of partial orderings intersection representation and one to one correspondence between linearity and maximality as a main result we obtain that it is virtually impossible to simultaneously preserve all these three properties in the fuzzy case if we do not require a one to one correspondence between linearity and maximality however we obtain that an implication based definition appears to constitute a sound compromise in particular if ukasiewicz type logics are considered

investigation of straightforward impedance eduction in the presence of shear flow abstract a straightforward impedance eduction method is proposed which combines prony s method with the pridmore brown equation to obtain the impedance of acoustic liners in the presence of shear flow particular attention is paid to the reported inconsistency problems associated with the boundary layer effect in the development of impedance eduction techniques a kind of flow insensitive acoustic liner is considered which is placed in a rectangular flow duct containing predominant grazing incidence mode three slip or no slip flow profiles are examined which are the parabola the one seventh power law and the uniform core flow with linear boundary layer a shear flow fem model is also set up to simulate the duct acoustic field the present impedance eduction method has been tested and validated using both the simulated and the published measured data it is shown that despite of their distinct boundary layers the selected flow profiles lead to essentially the same impedance results and also for the impedance eduction the strict consideration of no slip shear flow is very consistent with the use of the ingard myers boundary condition with the uniform flow assumption over the test conditions although not susceptible to the exact shape of flow profile the impedance eduction critically depends on the determination of the cross sectional average mach number the usual practice of approximately representing the duct flow with the midspan profile results in a slight overestimation of the average mach number in the two dimensional acoustic models and thus can considerably affect the accuracy of the impedance eduction to solve this problem the average flow profile is introduced to account for the actual three dimensional flow non uniformity in the rectangular duct it is further found that the effective mach number corresponding to a slightly modified average flow profile can be used to achieve a considerable collapse of the impedance spectra educed at different mach numbers based on the new evaluation of the influence of boundary layer profile the fem simulation using the straightforwardly educed impedance as the boundary condition can achieve an excellent agreement with the benchmark data for the complex sound pressure in the flow duct

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

special properties closures and interiors of crisp and fuzzy relations this paper offers optimal definitions of the special properties which relations on a set may possess and which combine to define the classically important types of preorders both kinds of orders tolerances and equivalences the useful concept of local properties is introduced the definitions are shown to unify in a highly convenient way fuzzy and crisp considerations the concepts of closure and interior of a relation with respect to a property are defined necessary and sufficient conditions for the full existence of specific closures and interiors are given and formulae for all of them are presented and validated a widely important area of application of the closure and interior operators is discussed finally fast algorithms are given for the computer calculation of the more troublesome closures and a standardized procedure for the analysis of local orders is presented

algorithms for the computation of t transitive closures we present two weight driven algorithms for the computation of the t transitive closure of a symmetric binary fuzzy relation on a finite universe x with cardinality n or equivalently of a symmetric n spl times n matrix with elements in 0 1 with t a triangular norm the first algorithm is proven to be valid for any triangular norm t whereas the second algorithm is shown to be valid when t is either the minimum operator or an archimedean triangular norm furthermore we investigate how these algorithms can be appropriately adapted to generate the t transitive closure of nonsymmetric binary fuzzy relations or matrices as well

an introduction to vague complemented ordered sets abstract the purpose of this paper is to introduce a theory of fuzzily defined complement operations on nonempty sets equipped with fuzzily defined ordering relations many valued equivalence relation based fuzzy ordering relations also called vague ordering relations provide a powerful and a comprehensive mathematical modelling of fuzzily defined partial ordering relations for this reason starting with a nonempty set x equipped with a many valued equivalence relation and a vague ordering relation a fuzzily defined complement operation called a vague complement operation on x will be formulated by means of the underling many valued equivalence relation and vague ordering relation because of the fact that the practical implementations of vague complement operations basically depend on their representation properties a considerable part of this paper is devoted to the representations of vague complement operations in addition to this the present paper provides various nontrivial examples for vague complements and introduces a many valued logical interpretation of quantum logic as a real application of vague complements

a plea for the use of lukasiewicz triplets in the definition of fuzzy preference structures i general argumentation abstract the generalization of the concept of a classical or crisp preference structure to that of a fuzzy preference structure expressing degrees of strict preference indifference and incomparability among a set of alternatives requires the choice of a de morgan triplet i e of a triangular norm and an involutive negator the resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences more specifically one of the degrees of strict preference indifference or incomparability should always be unconstrained to the preference modeller this intuitive requirement is violated when choosing a triangular norm without zero divisors since in that case fuzzy preference structures reduce to classical preference structures and hence none of the degrees can be freely assigned furthermore it is shown that the choice of a continuous non archimedean triangular norm having zero divisors is not compatible with our basic requirement the sets of degrees of strict preference indifference and incomparability in 0 1 are always bounded from above by a value strictly smaller than 1 these fundamental results imply that when working with a continuous triangular norm only archimedean ones having zero divisors are suitable candidates these arguments sufficiently support our plea for the use of ukasiewicz triplets in the definition of fuzzy preference structures

studies in fuzzy relations over fuzzy subsets in this paper a generalisation of fuzzy relations is introduced fuzzy relations are defined on fuzzy subsets properties like ordered reflexivity symmetry transitity transitive closures of such generalised relations and operations on them are discussed

mi algebras the existing arithmetics of fuzzy numbers usually stemming from the cut arithmetic do not preserve some of the important properties of the standard arithmetics of real numbers although one cannot expect a generalization of standard arithmetic to perfectly preserve all its properties the most important properties should be preserved at least in a weakened form we present a novel framework for arithmetics of extensional fuzzy numbers that more or less preserves all the important algebraic properties of the arithmetic of real numbers and thus seems to be an important seed for further investigations on this topic the suggested approach to arithmetics of extensional fuzzy numbers is demonstrated by many examples and besides its algebraic properties it is also shown to carry some desirable practical properties the investigation leads to novel algebraic structures mi algebras mi monoids mi groups mi fields that abstract the discussed properties the main idea of these structures is based on a set of pseudoidentities that complements the only commonly used identity element in a monoid group structure these pseudoidentities are elements that generalize the identity like behavior and allow us to weaken the standard forms of algebraic properties that are highly desirable to preserve at least in a weakened form appropriate properties of suggested mi structures are introduced and demonstrated by a number of practical examples

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

an adjoint based method for liner impedance eduction validation and numerical investigation abstract an inverse method for liner acoustic impedance eduction in a duct is presented the determination of adjoint equations allows to make explicit the directional derivatives of the objective function needed by the search algorithm direct and adjoint system of equations are spatially discretized by a discontinuous galerkin scheme the propagation model and the eduction process are successfully validated on nasa benchmark data extracted from literature in a no flow configuration objective functions written in terms of acoustic pressure and acoustic velocity are considered successively special attention is given to the possibilities offered by the method such as multiparameter identification with the study of a non homogeneous liner convergence history of the process is also examined revealing interesting properties for the velocity based objective function compared to the pressure based objective function in further developments the method is planned to be used with acoustic velocity fields obtained by laser doppler velocimetry

monoidal t norm based logic towards a logic for left continuous t norms hajek s bl logic is the fuzzy logic capturing the tautologies of continuous t norms and their residua in this paper we investigate a weaker logic mtl which is intended to cope with the tautologies of left continuous t norms and their residua the corresponding algebraic structures mtl algebras are defined and completeness of mtl with respect to linearly ordered mtl algebras is proved besides several schematic extensions of mtl are also considered as well as their corresponding predicate calculi

a non commutative and non idempotent theory of quantale sets in fuzzy set theory non idempotency arises when the conjunction is interpreted by arbitrary t norms there are many instances in mathematics where set theory ought to be non commutative and or non idempotent the purpose of this paper is to combine both ideas and to present a theory of non commutative and non idempotent quantale sets among other things standard concepts like fuzzy preorders and fuzzy equivalence relations will be exhibited as special cases more specifically the category q set is investigated where q is an arbitrary involutive quantale objects quantale sets are pairs consisting of a set and a q valued equality with a suitable symmetry axiom three important properties of q set are shown it is complete cocomplete and has the epi extremal mono factorization property its subcategory s q set of separated quantale sets is reflective in q set and shares the same fundamental properties with q set in particular s q set is also a complete and cocomplete epi extremal mono category the objects of the category q set are interesting categories in their own right two categorical frameworks for objects of q set are exhibited first it is shown that q valued equalities arise from q valued preorders with self adjoint extents by symmetrization which leaves q valued equalities invariant here sets with quantale preorders are shown to be b categories with base b being a specific quantaloid the second approach is based on involutive quantaloids a combination of two well known things quantaloids and ordered categories with involution in this context quantale sets are precisely symmetric b categories w r t appropriately chosen quantaloid b with involution further the cauchy completion preserves the symmetry axiom for a large class of involutive quantales which include quantic frames our non commutative generalization of frames and all left continuous t norms there exist at least two monads on q set the singleton monad and quasi singleton monad which are of special interest for fuzzy set theory the kleisli category associated with the singleton monad is the non commutative and non idempotent analogue of higgs topos while the eilenberg moore category of the quasi singleton monad permits the internalization of lukasiewicz negation as truth arrow finally an application of quantale sets to c algebras is given and the change of base is treated

conditional extensions of fuzzy preorders abstract the problem of embedding incomplete into complete relations has been an important topic of research in the context of crisp relations and their applications several variations of the acclaimed szpilrajn s theorem have been provided inclusive of the case when some order conditions between elements are imposed on the extension we extend the analysis of that topic by alcantud to the fuzzy case by appealing to generators to decompose fuzzy preference relations into strict preference and indifference relations we give general extension results for the corresponding concept of compatible extension of a fuzzy reflexive relation then we investigate the conditions under which compatible order extensions exist such that certain elements are connected by the asymmetric part resp and certain other elements by the symmetric part to respective elements with degree 1