Glad Deschrijver Bart Van Gasse Koen Ovaere Pavol Kr Ofer Arieli Micha Baczyski Etienne E. Kerre Chris Cornelis Martine De Cock Annelies Vroman Erich P. Klement Radko Mesiar Fuzzy Sets and Systems Information Sciences International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems IEEE Transactions on Fuzzy Systems International Journal of Approximate Reasoning Notes on Intuitionistic Fuzzy Sets

7 3 2001 14-22 file:///home/gdschrij/Fuzzy/2001/cartprod/NIFS-07-3-14-22.pdf

Journal of Fuzzy Mathematics 10 3 2002 711-724 file:///home/gdschrij/Fuzzy/2001/links/ifslinks2.pdf

8 1 2002 19-27 file:///home/gdschrij/Fuzzy/2002/ifops/ifsops.pdf

8 3 2002 1-10 file:///home/gdschrij/Fuzzy/2002/nifs/ift.pdf

MarianaNikolova NikolaiNikolov Advanced Studies in Contemporary Mathematics 4 2 2002 127-157

133 2 2003 1 16 227-235 Intuitionistic fuzzy set; Interval valued fuzzy set; Interval valued intuitionistic fuzzy set; Fuzzy set; Relationship between models Since Zadeh introduced fuzzy sets in 1965, a lot of new theories treating imprecision and uncertainty have been introduced. Some of these theories are extensions of fuzzy set theory, others try to handle imprecision and uncertainty in a different (better?) way. Kerre (Computational Intelligence in Theory and Practice, Physica-Verlag, Heidelberg, 2001, pp. 55-72) has given a summary of the links that exist between fuzzy sets and other mathematical models such as flou sets (Gentilhomme), two-fold fuzzy sets (Dubois and Prade) and L-fuzzy sets (Goguen). In this paper, we establish the relationships between intuitionistic fuzzy sets (Atanassov, VII ITKR's Session, Sofia, June 1983 (Deposed in Central Sci.-Techn. Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)), L-fuzzy sets (J. Math. Anal. Appl. 18 (1967) 145), interval-valued fuzzy sets (Sambuc, Ph.D. Thesis, University of Marseille, France, 1975), interval-valued intuitionistic fuzzy sets (Intuitionistic fuzzy set, Physica-Verlag, Heidelberg, New York, 1999). file:///home/gdschrij/Fuzzy/2003/links/ifslinks1final.pdf

136 3 2003 6 16 333-361 Fuzzy relation; Intuitionistic fuzzy set; Intuitionistic fuzzy relation; Triangular composition Fuzzy relations are able to model vagueness, in the sense that they provide the degree to which two objects are related to each other. However, they cannot model uncertainty: there is no means to attribute reliability information to the membership degrees. Intuitionistic fuzzy sets, as defined by Atanassov (Instuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999), give us a way to incorporate uncertainty in an additional degree. Intuitionistic fuzzy relations are intuitionistic fuzzy sets in a cartesian product of universes. One of the main concepts in relational calculus is the composition of two relations. Burillo and Bustince (Fuzzy Sets and Systems 78 (1996) 293; Soft Comput. 2 (1995) 5) have extended the sup-T composition of fuzzy relations to a composition of intuitionistic fuzzy relations. In this paper, we present an intuitionistic fuzzy version of the triangular compositions of Bandler and Kohout (in: P. Wang, S. Chang (Eds.), Theory and Application to Policy Analysis and Information Systems, Plenum Press, New York, 1980, p. 341) and the variants of these compositions given by De Baets and Kerre (Adv. Electron. Electron Phys. 89 (1994) 255). Some properties of these compositions are investigated: containment, convertibility, monotonicity, interaction with union and intersection. file:///home/gdschrij/Fuzzy/2003/relations/ifsrelation1final.pdf

Journal of Fuzzy Mathematics 11 3 2003 537-547

9 3 2003 11-21 file:///home/gdschrij/Fuzzy/2004/flins/nifse.tex

11 6 2003 691-709 file:///home/gdschrij/Fuzzy/2003/tnorms/iftnormrepres3final.pdf

35 1 2004 55-95 file:///home/gdschrij/Fuzzy/2003/implications/ijarfinal.pdf

12 1 2004 45-61 file:///home/gdschrij/Fuzzy/2004/iftnormrepres1/iftnormrepres1final.pdf

148 2 2004 243-262 file:///home/gdschrij/Fuzzy/2004/ifuninorm1/ifuninorm1final.pdf

153 2 2005 229-248 file:///home/gdschrij/Fuzzy/2005/agop/agop1final.pdf

Kuwait Journal of Science and Engineering 32 1 2005 17-38 file:///home/gdschrij/Fuzzy/2005/ifsrelation2/ifsrelation2e.pdf

13 3 2005 321-335 file:///home/gdschrij/Fuzzy/annelies/2005/ijufks/artikel-ijufks-final.pdf

New Mathematics and Natural Computation 1 2 2005 243-259 file:///home/gdschrij/Fuzzy/2005/LItn_ordinalsum1/LItn_ordinalsum1a.pdf

13 4 2005 453-464 file:///home/gdschrij/Fuzzy/2005/iftnormSM/iftnormSMfinal.pdf

Proceedings of the 8th European Conference on Symbolic and Quantitative Approaches to Reasoning With Uncertainty (ECSQARU'05), Lecture Notes in Computer Science 3571 2005 563-575 file:///home/gdschrij/Fuzzy/2004/flins/ecsqaru05b.ps.gz

Journal of Intelligent and Fuzzy Systems 16 4 2005 265-271 file:///home/gdschrij/Fuzzy/2005/kerre60/LIaddition2final.pdf

Journal of Electrical Engineering 56 12/s 2005 13-16 file:///home/gdschrij/Fuzzy/2005/bratislava/iscam2005LIadd.pdf

157 5 2006 3 1 622-627 Interval-valued fuzzy logic; Logical connectives; Algebraic structures; Representability; Fuzzy logics Among the various extensions to the common [0,1]-valued truth degrees of "traditional" fuzzy set theory, closed intervals of [0,1] stand out as a particularly appealing and promising choice for representing imperfect information, nicely accommodating and combining the facets of vagueness and uncertainty without paying too much in terms of computational complexity. From a logical point of view, due to the failure of the omnipresent prelinearity condition, the underlying algebraic structure L^I falls outside the mainstream of the research on formal fuzzy logics (including MV-, BL- and MTL-algebras), and consequently so far has received only marginal attention. This comparative lack of interest for interval-valued fuzzy logic has been further strengthened, perhaps, by taking for granted that its algebraic operations amount to a twofold application of corresponding operations on the unit interval. Abandoning that simplifying assumption, however, we may find that L^I reveals itself as a very rich and noteworthy structure allowing the construction of complex and surprisingly well-behaved logical systems. Reviewing the main advances on the algebraic characterization of logical operations on L^I, and relating these results to the familiar completeness questions (which remain as major challenges) for the associated formal fuzzy logics, this paper paves the way for a systematic study of interval-valued fuzzy logic in the narrow sense. file:///home/gdschrij/Fuzzy/2006/fuzzylogic/fuzzylogicfinal.pdf

New Mathematics and Natural Computation 2 1 2006 3 29-41 Formal fuzzy logic; Interval-valued fuzzy sets; Triangular norms; Residuated lattices Since it does not generate any MTL-algebra (prelinear residuated lattice), the lattice L^I of closed subintervals of [0,1] falls outside the mainstream of research on formal fuzzy logics. However, due to the intimate connection between logical connectives on L^I and those on [0,1], many relevant logical properties can still be maintained, sometimes in a slightly weaker form. In this paper, we focus on a broad class of parametrized t-norms on L^I. We derive their corresponding residual implicators, and examine commonly imposed logical properties. Importantly, we formally establish one-to-one correspondences between join-definability (respectively, weak divisibility) for t-norms of this class and strong join-definability (resp., divisibility) for their counterparts on [0,1]. file:///home/gdschrij/Fuzzy/bart/2006/IVFLPropGenClass/ivflpropgenclass.pdf

157 17 2006 9 1 2311-2327 Archimedean; Interval-valued fuzzy set; Limit property; t-norm; Strong Archimedean; Weakly Archimedean; Weak limit property In this paper the Archimedean property of t-norms on the lattice L^I is introduced, where L^I is the underlying lattice of interval-valued fuzzy set theory (Sambuc, 1975) and intuitionistic fuzzy set theory (Atanassov, 1983). Two additional variations on this property are also introduced: weakly and strong Archimedean property. We investigate the relationship with the (weak) limit property and obtain several necessary and sufficient conditions for a t-norm on L^I to be Archimedean. We discuss the Archimedean property for special classes of t-norms on L^I based on t-norms on the unit interval. file:///home/gdschrij/Fuzzy/2006/LItn_archimedean1/LItn_archimedean1final.pdf

Proceedings of the International Conference on Modeling Decisions for Artificial Intelligence (MDAI 2006), Lecture Notes in Computer Science 3885 2006 4 22-33 Bilattices; Preference modeling; Reasoning with uncertainty; Knowledge representation Many realistic decision aid problems are fraught with facets of ambiguity, uncertainty and conflict, which hamper the effectiveness of conventional and fuzzy preference modeling approaches, and command the use of more expressive representations. In the past, some authors have already identified Ginsberg's/Fitting's theory of bilattices as a naturally attractive candidate framework for representing uncertain and potentially conflicting preferences, yet none of the existing approaches addresses the real expressive power of bilattices, which lies hidden in their associated truth and knowledge orders. As a consequence, these approaches have to incorporate additional conventions and 'tricks' into their modus operandi, making the results unintuitive and/or tedious. By contrast, the aim of this paper is to demonstrate the potential of (rectangular) bilattices in encoding not just the problem statement, but also its generic solution strategy. file:///home/gdschrij/Fuzzy/2006/mdai/mdai2006lnai-3885-0022.pdf

Proceedings of the 5th International Conference on Rough Sets and Current Trends in Computing (RSCTC 2006), Lecture Notes in Computer Science 4259 2006 11 117-126 Triangle algebra; Residuated lattice; Interval-valued residuated lattice In this paper, we present triangle algebras: residuated lattices equipped with two modal, or approximation, operators and with a third angular point u, different from 0 (false) and 1 (true), intuitively denoting ignorance about a formula's truth value. We prove that these constructs, which bear a close relationship to several other algebraic structures including rough approximation spaces, provide an equational representation of interval-valued residuated lattices; as an important case in point, we consider L^I, the lattice of closed intervals of [0,1]. As we will argue, the representation by triangle algebras serves as a crucial stepping stone to the construction of formal interval-valued fuzzy logics, and in particular to the axiomatic formalization of residuated t-norm based logics on L^I, in a similar way as was done for formal fuzzy logics on the unit interval. file:///home/gdschrij/Fuzzy/bart/2006/RSTC2006/rsctc3107definitief.pdf

12 1 2006 30-37 file:///home/gdschrij/Fuzzy/2005/ifsoeafs/LItn_generator2.pdf

177 8 2007 4 15 1860-1866 Links between different models; Fuzzy set; Interval-valued fuzzy set; Intuitionistic fuzzy set; L-fuzzy set; Intuitionistic L-fuzzy set; Interval-valued intuitionistic fuzzy set; Probabilistic set; Soft set; Type 2 fuzzy set Intuitionistic fuzzy sets (Atanassov, 1983) are an extension of fuzzy set theory in which not only a membership degree is given, but also a non-membership degree which is more or less independent. Considering the increasing interest in intuitionistic fuzzy sets, it is useful to determine the position of intuitionistic fuzzy set theory in the framework of the different theories modelling imprecision. In this paper we discuss the mathematical relationship between intuitionistic fuzzy sets and other models of imprecision. file:///home/gdschrij/Fuzzy/2004/ifslinks3/INS7471final.pdf

15 1 2007 2 13-41 Bilattices; Intuitionistic fuzzy sets; Interval-valued fuzzy sets; Graded logical connectives; Preference modeling We present a family of algebraic structures, called rectangular bilattices, which serve as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs). These structures are useful on one hand to clarify the exact nature of the relationship between the above two common extensions of fuzzy sets, and on the other hand provide an intuitively attractive framework for the representation of uncertain and potentially conflicting information. We also provide these structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Application potential and intuitive appeal of the proposed framework are illustrated in the context of preference modeling. file:///home/gdschrij/Fuzzy/2007/blrectangle/S0218488507004352.pdf

15 2 2007 4 161-175 Bilattices; Bilattice-based squares and triangles; Negators; t-norms and t-conorms; Implicators; MV-algebras In this paper, Ginsberg's/Fitting's theory of bilattices, and in particular the associated constructs of bilattice-based squares and triangles, is introduced as an attractive framework for the representation of uncertain and potentially conflicting information, paralleling Goguen's L-fuzzy set theory. We recall some of the advantages of bilattice-based frameworks for handling fuzzy sets and systems, provide the related structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. file:///home/gdschrij/Fuzzy/2006/bltriangle/04142748.pdf

15 2 2007 4 222-237 Addition on L^I; Additive generator on L^I; Involutive negator; Multiplication on L^I; Multiplicative generator on L^I; Representable generator; t-norm on L^I In this paper, we introduce the notion of additive and multiplicative generator on L^I, which is the underlying lattice of interval-valued fuzzy set theory. A special class of generators on L^I is the class of representable generators which can be represented using generators on the unit interval. A representation of involutive negators on L^I by means of continuous additive generators which is similar to the representation of involutive negators on the unit interval is given. We also investigate additive and multiplicative generators of t-norms on L^I. file:///home/gdschrij/Fuzzy/2005/LItn_generator1/04142752.pdf

177 14 2007 7 15 2906-2924 t-norm on L^I; Arithmetic operators on L^I; Sum; Difference; Product; Quotient; Addition; Subtraction; Multiplication; Division We introduce the addition, subtraction, multiplication and division on L^I, where L^I is the underlying lattice of both interval-valued fuzzy set theory [R. Sambuc, Fonctions Φ-floues. Application à l’aide au diagnostic en pathologie thyroidienne, Ph.D. Thesis, Université de Marseille, France, 1975] and intuitionistic fuzzy set theory [K.T. Atanassov, Intuitionistic fuzzy sets, 1983, VII ITKR’s Session, Sofia (deposed in Central Sci. Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)]. We investigate some algebraic properties of these operators. We show that using these operators the pseudo-t-representable extensions of the Łukasiewicz t-norm and the product t-norm on the unit interval to L^I and some related operators can be written in a similar way as their counterparts on the unit interval. file:///home/gdschrij/Fuzzy/2007/LIaddition1/LIaddition1final.pdf

158 14 2007 6 16 1515-1534 Fuzzy number; System of fuzzy linear equations Buckley and Qu proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solutions of all systems of linear crisp equations formed by the α-levels are calculated. We propose a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. By observing the monotonicity of the parametric functions in each variable, i.e. each fuzzy coefficient in the system, we improve the algorithm by calculating less parametric functions and less evaluations of these parametric functions. We show that our algorithm is much more efficient than the method of Buckley and Qu. file:///home/gdschrij/Fuzzy/annelies/2006/improvedparametric/imprparamfinal.pdf

158 15 2007 7 1 1728-1750 Scalar cardinality; Interval-valued fuzzy set; Intuitionistic fuzzy set; Valuation property; Subadditivity; Complementarity rule; Cartesian product rule Wygralak has developed an axiomatic theory of scalar cardinalities of fuzzy sets with finite support which covers as particular cases all standard (historical) concepts of the scalar cardinality. In this paper we present a possible extension of this theory to interval-valued fuzzy set theory. We introduce the cardinality of interval-valued fuzzy sets as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞[. We study some properties of these cardinalities using t-norms, t-conorms and negations on the lattice L^I (the underlying lattice of interval-valued fuzzy set theory). In particular, the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule will be discussed. file:///home/gdschrij/Fuzzy/2007/LIcardinality1p1/FSS4977final.pdf

15 3 2007 6 370-384 Fuzzy numbers; Solving systems of fuzzy linear equations Buckley and Qu proposed a method to solve systems of linear fuzzy equations. Basically, in their method the solution of all systems of linear crisp equations formed by the alpha-levels are calculated. We propose in this paper a new method for solving systems of linear fuzzy equations based on a practical algorithm using parametric functions in which the variables are given by the fuzzy coefficients of the system. We show that our algorithm is much more efficient than the method of Buckley and Qu. file:///home/gdschrij/Fuzzy/annelies/2005/solveeqn/04231870.pdf

15 3 2007 6 345-361 Interval-valued fuzzy sets; L-fuzzy sets; Graded logical connectives; Representability Interval-valued fuzzy set theory is an increasingly popular extension of fuzzy set theory where traditional [0,1]-valued membership degrees are replaced by intervals in [0,1] that approximate the (unknown) membership degrees. To construct suitable graded logical connectives in this extended setting, it is both natural and appropriate to "reuse" ingredients from classical fuzzy set theory. In this paper, we compare different ways of representing operations on interval-valued fuzzy sets by corresponding operations on fuzzy sets, study their intuitive semantics, and relate them to an existing, purely order-theoretical approach. Our approach reveals, amongst others, that subtle differences in the representation method can have a major impact on the properties satisfied by the generated operations, and that contrary to popular perception, interval-valued fuzzy set theory hardly corresponds to a mere twofold application of fuzzy set theory. In this way, by making the mathematical machinery behind the interval-valued fuzzy set model fully transparent, we aim to foster new avenues for its exploitation by offering application developers a much more powerful and elaborate mathematical toolbox than existed before. file:///home/gdschrij/Fuzzy/2007/representability/S0218488507004716.pdf

159 9 2008 5 1042-1060 Formal logic; Interval-valued fuzzy set theory; Residuated lattices In this paper, we introduce triangle algebras: a variety of residuated lattices equipped with approximation operators, and with a third angular point u, different from 0 and 1. We show that these algebras serve as an equational representation of interval-valued residuated lattices (IVRLs). Furthermore, we present Triangle Logic (TL), a system of many-valued logic capturing the tautologies of IVRLs. Triangle algebras are used to cast the essence of using closed intervals of L as truth values into a set of appropriate logical axioms. Our results constitute a crucial first step towards solving an important research challenge: the axiomatic formalization of residuated t-norm based logics on L^I, the lattice of closed intervals of [0,1], in a similar way as was done for formal fuzzy logics on the unit interval. file:///home/gdschrij/Fuzzy/bart/2007/TriangleLogic/fssartikeltrianglealgenlogfinal.pdf

159 13 2008 7 1597-1618 Lattice of closed intervals; Interval-valued fuzzy set; Triangular norm; Representation; Join-morphism; Residuation principle In this paper we consider the lattice L^I which has the closed subintervals of a complete lattice as elements. We give a representation theorem of t-norms on this lattice for which the partial mappings are join-morphisms in terms of t-norms on the underlying lattice. In fuzzy logic, t-norms which satisfy the residuation principle play an important role. Using our representation theorem, we represent t-norms on L^I which satisfy the residuation principle and two border conditions in terms of t-norms on the underlying lattice. In the case that the underlying lattice of L^I is the unit interval, we obtain characterizations of continuous t-norms which are natural extensions of t-norms on the unit interval and which have join-morphisms as partial mappings or alternatively satisfy the residuation principle. file:///home/gdschrij/Fuzzy/2007/LItn_repres5/FSS5151final.pdf

International Journal of Computational Intelligence Systems 1 3 2008 8 248-261 Fuzzy number; System of linear fuzzy equations; Symmetric matrix; Parametric function A method to solve linear fuzzy equations with a symmetric matrix is proposed. Ignoring the symmetry leads to an overestimation of the solution. Our method to find the solution of a system of linear fuzzy equations takes the symmetry of the matrix into account and is based on parametric functions. It is a practical algorithm using parametric functions in which the variables are given by elements of the support of the fuzzy coefficients of the system. file:///home/gdschrij/Fuzzy/annelies/2007/iske/extISKEfinal.pdf

49 2 2008 10 478-487 Interval-valued fuzzy set theory; Residuated lattices; Triangle algebras As is well-known, residuated lattices (RLs) on the unit interval correspond to left-continuous t-norms. Thus far, a similar characterization has not been found for RLs on the set of intervals of [0,1], or more generally, of a bounded lattice L. In this paper, we show that the open problem can be solved if it is restricted, making only a few simple and intuitive assumptions, to the class of interval-valued residuated lattices (IVRLs).More specifically, we derive a full characterization of product and implication in IVRLs in terms of their counterparts on the base RL. To this aim, we use triangle algebras, a recently introduced variety of RLs that serves as an equational representation of IVRLs. file:///home/gdschrij/Fuzzy/bart/2008/prodandimplintrianglealg/IJA_7143final.pdf

160 6 2009 3 778-801 Interval-valued fuzzy set theory; t-Norm; (weakly) Archimedean; (weakly) Nilpotent; Strict In this paper we first give characterizations of the class of continuous t-norms on L^I (where L^I is the lattice of closed subintervals of the unit interval) which satisfy the residuation principle and which are a natural extension of a t-norm on the unit interval and which satisfy one of the following conditions: the negation generated by their residual implication is involutive; they are (weakly) Archimedean; they are (weakly) nilpotent. We fully characterize the class of continuous t-norms on L^I which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are weakly Archimedean. We construct a separate representation for the t-norms in this class which are weakly nilpotent and for those which are not weakly nilpotent. Finally we give a characterization of the continuous t-norms on L^I which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are strict. file:///home/gdschrij/Fuzzy/2008/LItn_archimedean2/FSS_5383final.pdf

Fuzzy Information and Engineering 1 1 2009 3 27-36 Fuzzy finite element method; Fuzzy ordering; Fuzzy decision making In this paper we present a new approach to handle uncertainty in the Finite Element Method. As this technique is widely used to tackle real-life design problems, it is also very prone to parameter-uncertainty. It is hard to make a good decision regarding design optimization if no claim can be made with respect to the outcome of the simulation. We propose an approach that combines several techniques in order to offer a total order on the possible design choices, taking the inherent fuzziness into account. Additionally we propose a more efficient ordering procedure to build a total order on fuzzy numbers. file:///home/gdschrij/Fuzzy/koen/2007/paper4-2.pdf

179 6 2009 3 717-728 Interval-valued fuzzy set theory; Residuated lattices; Formal logic Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct triangle logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRLs.In this paper, we prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices.As a consequence, we are able to prove the pseudo-chain completeness of pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of monoidal T-norm based logic (MTL).This result also provides a better insight in the structure of triangle algebras; it enables us, amongst others, to prove properties of pseudo-prelinear triangle algebras more easily. It is known that there is a one-to-one correspondence between triangle algebras and couples (L,alpha), in which L is a residuated lattice and alpha an element in that residuated lattice. We give a schematic overview of some properties of pseudo-prelinear triangle algebras (and a number of others that can be imposed on a triangle algebra), and the according necessary and sufficient conditions on L and alpha. file:///home/gdschrij/Fuzzy/bart/2008/filterspseudochaincompleteness/INS_8139final.pdf

DonalO'Regan RezaSaadati S. MansourVaezpour Chaos, Solitons and Fractals 42 1 2009 10 40-45 L-fuzzy Euclidean normed space; Compactness In this paper, L-fuzzy Euclidean normed spaces are defined and compactness in these spaces is considered. file:///home/gdschrij/Fuzzy/2009/ifnormedspace/Chaos6514_final.pdf

PatriciaVictor 17 3 2009 6 630-640 Constrained fuzzy arithmetics; Real-time computations; Fuzzy numbers Klir introduced constrained fuzzy arithmetics (CFA) as a solution to the unnecessary precision loss when dealing with fuzzy quantities which represent linguistic variables. Since then, some attempts have been made to make CFA efficient, but none of these solutions is suitable for real-time applications. In this paper, we will propose a new CFA algorithm that can be used in such environments. file:///home/gdschrij/Fuzzy/annelies/2006/constrained/TFS-04505355final.pdf

160 21 2009 11 3080-3102 Interval-valued fuzzy set; Lukasiewicz t-norm; Product t-norm; Frank t-norm; Schweizer-Sklar t-norm; Yager t-norm; Arithmetic operators In this paper we study extensions of the arithmetic operators +, -, x, / to the lattice L^I of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on L^I which can be generated using these operators; these classes provide natural extensions of the Lukasiewicz, product, Frank, Schweizer-Sklar and Yager t-norms to L^I. file:///home/gdschrij/Fuzzy/2009/LIaddition3/LIaddition3final.pdf

180 16 2010 8 3006-3020 Filters; Interval-valued structures; Residuated lattices An important concept in the theory of residuated lattices and other algebraic structures used for formal fuzzy logic, is that of a filter. Filters can be used, amongst others, to define congruence relations. Specific kinds of filters include Boolean filters and prime filters.In this paper, we define several different filters of residuated lattices and triangle algebras and examine their mutual dependencies and connections. Triangle algebras characterize interval-valued residuated lattices. file:///home/gdschrij/Fuzzy/bart/2010/IVRLfilters/ivrlfilters_final.pdf

PeterSussner MikeNachtegael TomMlange EstevoEsmi Journal of Mathematical Imaging and Vision 2011 submitted Mathematical morphology; Complete lattice; L-fuzzy sets; Interval-valued fuzzy sets; Atanassov's intuitionistic fuzzy sets; L-fuzzy mathematical morphology; L-fuzzy connectives; Inclusion measure; Duality; Negation; Adjunction Abstract Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted.The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing. file:///home/gdschrij/Fuzzy/2010/IVFSmorphology/JMIV-698-2011-02-02.pdf

in press file:///home/gdschrij/Fuzzy/2010/LItn_infmorph/LItn_infmorphb.pdf

submitted file:///home/gdschrij/Fuzzy/2011/LIuninorm2/LIuninorm2.pdf

submitted file:///home/gdschrij/Fuzzy/bart/2009/standardcompleteness/standardcompletenessrevision.pdf

in preparation

in preparation

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in preparation

Ph. D. Thesis

Ghent University, Belgium

2004 11 9 in Dutch Fuzzy sets (Zadeh, 1965) are an extension of classical sets which allow a gradual transition between "belonging to" and "not belonging to" when determining the membership of an object in a set. Intuitionistic fuzzy sets (Atanassov, 1983) form an extension of fuzzy sets: they assign to each element of the universe not only a membership degree but also a non-membership degree satisfying some border condition. Intuitionistic fuzzy sets are equivalent to interval-valued fuzzy sets (Sambuc, 1975) which map every element of the universe on a closed subinterval of [0,1] giving an approximation of the "real" (but unknown) membership degree.In this work the properties of the mathematical operators and structures that play an important role in intuitionistic fuzzy set theory are discussed. We introduce the lattice L* and show that intuitionistic fuzzy sets are L-fuzzy sets (Goguen, 1967) w.r.t. the lattice L*. We extend the notions of t-norm, t-conorm, negator, implicator and uninorm to L*. We prove a representation of involutive negators on L* in terms of negators on [0,1], and similarly for increasing bijections on L* with increasing inverse. An example of a continuous, increasing permutation of L* which does not have an increasing inverse (contrary to bijections on [0,1]) is constructed. We show that not all t-norms and t-conorms on L* are defined componentwise in terms of t-norms and t-conorms on [0,1], a counterexample is given by the Lukasiewicz t-norm on L*. We introduce the residuation principle for t-norms on L* and investigate some related properties. We investigate the properties of t-norms, t-conorms and implicators on L* and prove several representation theorems which involve the Lukasiewicz t-norm on L*. Uninorms form a generalization of t-norms and t-conorms. We study some properties of uninorms on L*, representable uninorms on L* and implicators generated by uninorms on L*. We discuss furthermore some properties of the cartesian product of intuitionistic fuzzy sets. Finally, we extend the sup-T composition to intuitionistic fuzzy relations, as well as the triangular compositions of Bandler and Kohout and their improved versions of De Baets and Kerre and investigate their properties and alpha-cuts. file:///home/gdschrij/Fuzzy/2004/phd/phd/phd.pdf Applied Computational Intelligence DaRuan PierreD'hondt MikeNachtegael World Scientific 2004 57-64 file:///home/gdschrij/Fuzzy/2004/flins/flins04finalc.pdf Logical, Algebraic, Analytic, and Probabilistic Aspects of Triangular Norms Elsevier

Amsterdam

2005 231-259 file:///home/gdschrij/Fuzzy/2004/iftnormrepres4/08DeschrijverKerre_Final.pdf Issues in the Representation and Processing of Uncertain and Imprecise Information. Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets, and Related Topics Krassimir T.Atanassov JanuszKacprzyk MaciejKrawczak EulaliaSzmidt Akademicka Oficyna Wydawnicza EXIT 2005 83-101 file:///home/gdschrij/Fuzzy/2005/ifs2005/ifs2005luktnorm.pdf Applied Artificial Intelligence DaRuan PierreD'hondt Paolo F.Fantoni MikeNachtegael World Scientific 2006 43-50 Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, Studies in Fuzziness and Soft Computing HumbertoBustince FranciscoHerrera JavierMontero Springer-Verlag 2008 183-203 file:///home/gdschrij/Fuzzy/2006/LIagop_overview/LIagop_overviewc.pdf Computational Intelligence in Decision and Control, Proceedings of the 8th International FLINS Conference (FLINS 2008) DaRuan JuanMontero JieLu LuisMartnez PierreD'hondt World Scientific

Madrid (Spain)

2008 223-228 Interval-valued fuzzy set theory; Residuated lattices; Formal logic Triangle Logic is a formal fuzzy logic with intervals as truth values. Its construction is based on triangle algebras: equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which were called interval-valued residuated lattices (IVRLs). We prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices. Using this result, we prove an analogue of the chain completeness of MTL for Pseudo-prelinear Triangle Logic. It also enables us to prove properties of pseudo-prelinear triangle algebras more easily. We give some examples. file:///home/gdschrij/Fuzzy/bart/2008/flins2008.pdf MikeNachtegael StevenSchockaert YunShi Studies in Fuzziness and Soft Computing 261 Springer-Verlag

Berlin, Heidelberg

14 10 2010 10.1007/978-3-642-16629-7 978-3-642-16628-0 336 35 Years of Fuzzy Set Theory: Celebratory Volume Dedicated to the Retirement of Etienne E. Kerre MikeNachtegael StevenSchockaert YunShi Springer-Verlag

Berlin, Heidelberg

2010 57-82 Interval-valued fuzzy set theory; Residuated lattices; Formal logic In this chapter, we present a propositional calculus for several interval-valued fuzzy logics, i.e., logics having intervals as truth values. More precisely, the truth values are preferably subintervals of the unit interval. The idea behind it is that such an interval can model imprecise information. To compute the truth values of 'p implies q' and 'p and q', given the truth values of p and q, we use operations from residuated lattices. This truth-functional approach is similar to the methods developed for the well-studied fuzzy logics. Although the interpretation of the intervals as truth values expressing some kind of imprecision is a bit problematic, the purely mathematical study of the properties of interval-valued fuzzy logics and their algebraic semantics can be done without any problem. This study is the focus of this chapter. file:///home/gdschrij/Fuzzy/2010/kerre65/VanGasse_final.pdf Proceedings of the Sixth ESSLLI Student Session K.Striegnitz 2001 83-94 file:///home/gdschrij/Fuzzy/2001/esslli/chrisglad.tex 6th Joint Conference on Information Sciences (JCIS 2002) Research Triangle Park Durham, North Carolina, USA Special Session on Intuitionistic Fuzzy Sets: Theory and Applications H. J.Caulfield S.Chen H.Chen R.Duro V.Honaver M.Lu M. G.Romay T. K.Shih D.Ventura P. P.Wang Y.Yang 2002 3 8-13 105-108 file:///home/gdschrij/Fuzzy/2002/durham/ifimplicators.pdf MarianaNikolova NikolaiNikolov 6th Joint Conference on Information Sciences (JCIS 2002) H. JohnCaulfield Shu-HengChen Heng-DaChen Richard J.Duro VasantHonaver MiLu Manuel GranaRomay Timothy K.Shih DanVentura Paul P.Wang YuanyuanYang 2002 117-120 file:///home/gdschrij/Fuzzy/2002/durham/survey.ps 9th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2002) Annecy, France 2002 7 1-5 1839-1844 file:///home/gdschrij/Fuzzy/2002/ipmu/ipmu2002final.pdf 1st International IEEE Symposium Intelligent Systems: Methodology, Models, Applications in Emerging Technologies 1 2002 340-345 file:///home/gdschrij/Fuzzy/2002/varna/varnaf.pdf International Conference on Fuzzy Information Processing: Theory and Applications (FIP 2003) Beijing, China 2003 3 1-4 7-12 file:///home/gdschrij/Fuzzy/2003/relations/ifsrelation3.tex 24th Linz Seminar on Fuzzy Set Theory: Triangular Norms and Related Operators in Many-Valued Logics Linz, Austria 2003 2 4-8 51-53 file:///home/gdschrij/Fuzzy/2003/linz/iftnormlinz2003.tex 3rd International Conference in Fuzzy Logic and Technology (EUSFLAT 2003) Zittau, Germany Special Session on Intuitionistic Fuzzy Sets and Related Concepts M.Wagenknecht R.Hampel 2003 9 10-12 164-167 file:///home/gdschrij/Fuzzy/2003/eusflat/iftnormeusflat2003.tex Krassimir T.Atanassov Daniela A.Orozova Scientific Conference Contemporary Technologies - 03 2003 32-39 file:///home/gdschrij/Fuzzy/2003/burgas/cek2.tex 10th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2004) Perugia, Italy 2004 7 4-9 1111-1116 file:///home/gdschrij/Fuzzy/2004/ipmu/ipmu2004agop2b.ps 10th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2004) 2004 1389-1396 file:///home/gdschrij/Fuzzy/2004/ipmu/ipmu2004ifs2.pdf 5th International Conference on Recent Advances in Soft Computing (RASC 2004) 2004 342-347 7th Conference in Applied Mathematics (ISCAM 2005) Bratislava, Slovakia 2005 4 15-16 12 file:///home/gdschrij/Fuzzy/2005/bratislava/iscam2005LIaddabstract.pdf 8th Joint Conference on Information Sciences (JCIS 2005) 2005 207-210 file:///home/gdschrij/Fuzzy/annelies/2005/durham2005/parametricfunctions.ps Workshop on Intuitionistic Fuzzy Sets and Other Extensions/Applications of Fuzzy Sets Gent, Belgium 2005 9 2 2-3 file:///home/gdschrij/Fuzzy/2005/ifsoeafs/ifsoeafs2005agop.pdf 4th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2005) Barcelona, Spain 2005 9 7-9 253-258 file:///home/gdschrij/Fuzzy/2005/eusflat/eusflat2005agopb.pdf 8th International Conference on Fuzzy Set Theory and Applications (FSTA 2006) EvaDrobn FerdinandChovanec 2006 67 file:///home/gdschrij/Fuzzy/2006/fsta/fsta2006card.ps 8th International Conference on Fuzzy Set Theory and Applications (FSTA 2006) EvaDrobn FerdinandChovanec

Liptovsk Jn (Slovak Republic)

2006 110 file:///home/gdschrij/Fuzzy/2006/fsta/fsta2006ivflc.ps 11th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006) Paris, France 2006 7 2-7 580-586 file:///home/gdschrij/Fuzzy/2006/ipmu/ipmu2006agopb.pdf 11th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006) 2006 2348-2354 file:///home/gdschrij/Fuzzy/2006/ipmu/ipmu2006ifsb.pdf Studia Logica Conference towards Mathematical Philosophy

Torun (Poland)

2006 143-144 file:///home/gdschrij/Fuzzy/bart/2006/StudiaLogica2006/studialogicaabstracttrianglelogic.pdf Workshop Intelligent Systems for Data Mining and Information Processing: Methods and Applications

Gent

2006 26 file:///home/gdschrij/Fuzzy/bart/2006/Bart.VanGasse@UGent.be(NAFIPS2006)b.pdf M.De Munck D.Moens D.Vandepitte ISMA 2006 International Conference on Noise and Vibration Engineering 2006 4227-4241 file:///home/gdschrij/Fuzzy/annelies/2006/isma2006/isma2006-4227_4242.pdf 4th International Summer School on Aggregation Operators (AGOP 2007) Gent, Belgium BernardDe Baets Koen C.Maes Academia Press

Gent

2007 7 9-14 185-191 Interval-valued fuzzy set; t-norm; Arithmetic operators In this paper we study extensions of the arithmetic operators +, -, *, / to the lattice L^I of closed subintervals of the unit interval. Starting from a minimal set of axioms that these operators must fulfill, we investigate which properties they satisfy. We also investigate some classes of t-norms on L^I which can be generated using these operators. file:///home/gdschrij/Fuzzy/2007/agop/agop2007b.pdf 5th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2007) Ostrava, Czech Republic Martintpnika VilmNovk UlrichBodenhofer University of Ostrava 2007 9 11-14 53-60 Interval-valued fuzzy set; Intuitionistic fuzzy set; t-norm; Archimedean; Nilpotent; Strict; Representation In this paper the Archimedean property and the nilpotency of t-norms on the lattice L^I is investigated, where L^I is the underlying lattice of interval-valued fuzzy set theory (Sambuc, 1975) and intuitionistic fuzzy set theory (Atanassov, 1983). We give some characterizations of continuous t-norms on L^I which satisfy the residuation principle, T(D,D) is a subset of D, the Archimedean property and nilpotency. file:///home/gdschrij/Fuzzy/2007/eusflat/eusflat2007agopb.pdf 5th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT 2007) Martintpnika VilmNovk UlrichBodenhofer University of Ostrava 2007 13-19 Fuzzy logic; Interval-valued fuzzy set theory; Residuated lattices Fuzzy formal logics were introduced in order to handle graded truth values instead of only 'true' and 'false'. A wide range of such logics were introduced successfully, like Monoidal T-norm based Logic, Basic Logic, Godel Logic, Lukasiewicz Logic etc. However, in general, fuzzy set theory is not only concerned with vagueness, but also with uncertainty. A possible solution is to use intervals instead of real numbers as membership values. In this paper, we present an approach with triangle algebras, which are algebraic characterizations of interval-valued residuated lattices. The variety of these structures corresponds in a sound and complete way to a logic that we introduce, called Triangle Logic (in the same way as, e.g., BL-algebras and Basic Logic). We will show that this truth-functional approach, along with the residuation principle, has some consequences that seem to obstruct an easy and proper interpretation for the semantics of Triangle Logic. file:///home/gdschrij/Fuzzy/bart/2007/eusflat/EUSFLAT2007-bartvangasse2.pdf Proceedings of the International Conference on Intelligent Systems and Knowledge Engineering (ISKE 2007) TianruiLi YangXu DaRuan 2007 1170 Systems of linear fuzzy equations; Symmetry; Parametric functions A method to solve linear fuzzy equations with a symmetric matrix is proposed. Ignoring the symmetry leads to an overestimation of the solution. Our method to find the solution of a system of linear fuzzy equations takes the symmetry of the matrix into account and is based on parametric functions. It is a practical algorithm using parametric functions in which the variables are given by elements of the support of the fuzzy coefficients of the system. 9th International Conference on Fuzzy Set Theory and Applications (FSTA 2008) 2008 73 file:///home/gdschrij/Fuzzy/2008/fsta/abstractFSTA2008GDPK2008.pdf 9th International Conference on Fuzzy Set Theory and Applications (FSTA 2008)

Liptovsk Jn (Slovak Republic)

2008 126 file:///home/gdschrij/Fuzzy/bart/2008/BartVanGasse_abstractFSTA2008.pdf Leuven Symposium on Applied Mechanics in Engineering (LSAME 2008) 2008 881-890 Systems of linear fuzzy equations; Symmetry; Parametric functions A method to solve linear fuzzy equations with a symmetric matrix is proposed. Ignoring the symmetry leads to an overestimation of the solution. Our method to find the solution of a system of linear fuzzy equations takes the symmetry of the matrix into account and is based on parametric functions. It is a practical algorithm using parametric functions in which the variables are given by elements of the support of the fuzzy coefficients of the system. file:///home/gdschrij/Fuzzy/annelies/2007/LSAME.08-NDM.08/LSAME2008.pdf 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2008) Mlaga, Spain LuisMagdalena ManuelOjeda-Aciego Jos LuisVerdegay 2008 6 22-27 1336-1343 file:///home/gdschrij/Fuzzy/2008/ipmu/ipmu2008agopb.pdf 12th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2008) LuisMagdalena ManuelOjeda-Aciego Jos LuisVerdegay 2008 1640-1647 file:///home/gdschrij/Fuzzy/2008/ipmu/ipmu2008ifsb.pdf BLAST 2008

University of Denver, Colorado (USA)

2008 6-7 file:///home/gdschrij/Fuzzy/bart/2008/abstractblast2008.pdf 6th Conference of the European Society for Fuzzy Logic and Technology/13th International Fuzzy Systems Association World Congress (EUSFLAT/IFSA 2009) Lisbon, Portugal Joo PauloCarvalho DidierDubois UzayKaymak Joo M. C.Sousa Technical University of Lisbon 2009 7 20-24 184-188 Interval-valued fuzzy set; Intuitionistic fuzzy set; Uninorm; Conjunctive; Disjunctive Uninorms are a generalization of t-norms and t-conorms for which the neutral element is an element of [0,1] which is not necessarily equal to 0 (as for t-norms) or 1 (as for t-conorms). Uninorms on the unit interval are either conjunctive or disjunctive, i.e. they aggregate the pair (0,1) to either 0 or 1. In real-life applications, this kind of aggregation may be counter-intuitive. Atanassov's intuitionistic fuzzy set theory is an extension of fuzzy set theory which allows to model uncertainty about the membership degrees. In Atanassov's intuitionistic fuzzy set theory there exist uninorms which are neither conjunctive nor disjunctive. In this paper we study such uninorms more deeply and we investigate the structure of these uninorms. We also give several examples of uninorms which are neither conjunctive nor disjunctive. file:///home/gdschrij/Fuzzy/2009/eusflat/eusflat2009ifsfinal.pdf Eurofuse Workshop Preference Modelling and Decision Analysis (EUROFUSE 2009) Pamplona, Spain BernardDe Baets JanosFodor PedroBurillo HumbertoBustince Public University of Navarra 2009 9 16-18 Interval-valued fuzzy set; Atanassov's Intuitionistic fuzzy set; Arithmetic operator In this paper we study arithmetic operators in interval-valued fuzzy set theory in an axiomatic way. We investigate several properties and we give a characterization of the class of arithmetic operators that act componentwisely if one of their arguments is a trivial interval (a singleton). file:///home/gdschrij/Fuzzy/2009/eurofuse/eurofuse2009ifsb.pdf 13th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2010) Dortmund, Germany Communications in Computer and Information Science EykeHllermeier RudolfKruse FrankHoffmann Springer-Verlag

Berlin Heidelberg

81 2010 6 28-31 412-419 interval-valued fuzzy set; t-norm; residuated lattice When interval-valued fuzzy sets are used to deal with uncertainty, using a single t-norm to model conjunction and a single implication leads to counterintuitive results. Therefore it is necessary to look beyond the traditional structures such as residuated lattices, and to investigate whether these structures can be extended using more than one product and implication. In this paper we will investigate under which conditions a number of properties that are valid in a residuated lattice are still valid when different products and implications are used. /home/gdschrij/Fuzzy/2010/ipmu/00810412.pdf BenjaminBedregal HumbertoBustince JavierFernandez 13th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2010) Dortmund, Germany Theory and applications of intuitionistic fuzzy sets Communications in Computer and Information Science EykeHllermeier RudolfKruse FrankHoffmann Springer-Verlag

Berlin Heidelberg

80 2010 6 28-31 591-600 Atanassov's intuitionistic fuzzy sets; fuzzy negation; contractive; Lipschitz property In this work we consider the concept of contractive Atanassov's intuitionistic mappings. In particular, we show that with our definitions, the only strict (strong) contractive negation is the standard one. /home/gdschrij/Fuzzy/2010/ipmu/00800591.pdf BenjaminBedregal HumbertoBustince JavierFernandez 2010 IEEE World Congress on Computational Intelligence (WCCI 2010) Barcelona, Spain IEEE

New York

2010 7 18-23 Interval-valued fuzzy sets; fuzzy negation; contractive; Lipschitz property In this work we consider the concept of contractive interval-valued fuzzy negation, as a negation such that it does not increase the length or amplitude of an interval. We relate this to the concept of Lipschitz function. In particular, we prove that the only strict (strong) contractive interval-valued fuzzy negation is the one generated from the standard (Zadeh's) negation. /home/gdschrij/Fuzzy/2010/WCCI2701def.pdf 6th International Conference on Fuzzy Logic and Intelligent Technologies in Nuclear Science (FLINS 2006) Blankenberge, Belgium Special Session on Intuitionistic Fuzzy Sets 2004 9 1-3