extending fuzzy logics with many hedges abstract fuzzy logic aims at modeling logical reasoning with vague or imprecise statements which may contain linguistic hedges in fact many hedges e g very highly rather and slightly can be used simultaneously to express different levels of emphasis moreover each hedge might have a dual one e g slightly can be seen as a dual hedge of very thus it is necessary to extend systems of fuzzy logic with multiple hedges this work proposes two axiomatizations for multiple hedges as an expansion of a core fuzzy logic in one axiomatization hedges do not have any dual one while in the other each hedge can have its own dual one it is shown that the proposed logics not only cover a large class of hedge functions but also have all completeness properties as the underlying logic w r t the class of their chains as well as distinguished subclasses of their chains including standard completeness the axiomatizations are also extended to the first order level furthermore we present a method to build linguistic fuzzy logics based on the axiomatizations and a hedge algebra whose corresponding algebras use a linguistic truth domain taken from the hedge algebra for representing and reasoning with linguistically expressed human knowledge where truth values of vague sentences are given in linguistic terms

mtl algebras arising from partially ordered groups after dropping the implication like operation and reversing the order an mtl algebra becomes a partially ordered structure l 0 1 based on the single addition like operation furthermore may be restricted to a partial but cancellative addition without loss of information we deal in this paper with the case that the resulting partial algebra l 0 1 is embeddable into the positive cone of a partially ordered group it turns out that most of the examples of mtl algebras known at present can be represented in this way so the systematisation of a considerable class of mtl algebras is achieved and the methods of their construction are brought onto a common line

uninorm logic with the n potency axiom this paper investigates a class of extensions c nul of uninorm logic ul which is obtained by adding the n potency axiom to ul the main result of this paper is that by generalizing jenei and montagna style approach for proving standard completeness for monoidal t norm based logic mtl we obtain the standard completeness for c nul in addition we present the suitable analytic hypersequent calculi for c nul

which logic is the real fuzzy logic this paper is a contribution to the discussion of the problem whether there is a fuzzy logic that can be considered as the real fuzzy logic we give reasons for taking imtl bl email protected and ev l fuzzy logic with evaluated syntax as those fuzzy logics that should be indeed taken as the real fuzzy logics

on ordinal sums of triangular norms on bounded lattices ordinal sums have been introduced in many different contexts e g for posets semigroups t norms copulas aggregation operators or quite recently for hoops in this contribution we focus on ordinal sums of t norms acting on some bounded lattice which is not necessarily a chain or an ordinal sum of posets necessary and sufficient conditions are provided for an ordinal sum operation yielding again a t norm on some bounded lattice whereas the operation is determined by an arbitrary selection of subintervals as carriers for arbitrary summand t norms by such also the structure of the underlying bounded lattice is investigated further it is shown that up to trivial cases there are no ordinal sum t norms on product lattices in general

integral equations of fields on a rotating black hole it is known that the radial equation of the massless fields with spin around kerr black holes cannot be solved by special functions recently the analytic solution was obtained by use of the expansion in terms of the special functions and various astrophysical application have been discussed it was pointed out that the coefficients of the expansion by the confluent hypergeometric functions are identical to those of the expansion by the hypergeometric functions we explain the reason of this fact by using the integral equations of the radial equation it is shown that the kernel of the equation can be written by the product of confluent hypergeometric functions the integral equation transforms the expansion in terms of the confluent hypergeometric functions to that of the hypergeometric functions and vice versa which explains the reason why the expansion coefficients are universal

integral relations for solutions of the confluent heun equation firstly we construct kernels for integral relations among solutions of the confluent heun equation che additional kernels are systematically generated by applying substitutions of variables secondly we establish integral relations between known solutions of the che that are power series and solutions that are series of special functions thirdly by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation a particular che from this solution we construct new series solutions of the general che and show that these are suitable for solving the radial part of the two center problem in quantum mechanics

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

the quaternion formalism for mobius groups in four or fewer dimensions abstract the mobius group of r n defines n dimensional inversive geometry this geometry can serve as an alternative to projective geometry in providing a common foundation for spherical euclidean and hyperbolic geometry accordingly the mobius group plays an important role in geometry and topology the modern emphasis on low dimensional topology makes it timely to discuss a useful quaternion formalism for the mobius groups in four or fewer dimensions the present account is self contained it begins with the representation of quaternions by 2 x 2 matrices of complex numbers it discusses 2 x 2 matrices of quaternions and how a suitably normalized subgroup of these matrices extended by a certain involution related to sense reversal is 2 1 homomorphic to the mobius group acting on r 4 it provides details of this action and the relation of this action to various models of the classical geometries in higher dimensions n 5 the best description of the mobius group is probably by means of n 2 n 2 lorentz matrices in the lower dimensions covered by the quaternion formalism this alternative lorentz formalism is a source of interesting homomorphisms a sampling of these homomorphisms is computed explicitly both for intrinsic interest and for an illustration of the ease with which one can handle the quaternion formalism

solution of a system of linear equations with fuzzy numbers in this paper the interval nature of fuzzy numbers is revealed by showing that many interesting results from classical interval analysis transfer also into the fuzzy case the paper deals with a solution of a fuzzy interval system of linear equations i e a system in which fuzzy intervals numbers appear instead of crisp numbers

a note on axiomatizations of pavelka style complete fuzzy logics pavelka style completeness a property relating degrees of provability and truth was previously studied mainly in the context of logics with continuous connectives it is known that in some other logics one can use infinitary deduction rule s to retain this form of completeness the present paper offers a systematic study of this idea for fuzzy logics which expand mtl and are given by a fixed standard algebra we explore the structure of the class of all reasonable expansions of any such logic by rational truth constants and for several prominent cases provide axiomatizations of particular expansions enjoying the pavelka style completeness

quaternionic modular groups abstract matrices whose entries belong to certain rings of algebraic integers are known to be associated with discrete groups of transformations of inversive n space or hyperbolic n 1 space h n 1 in particular groups operating in the hyperbolic plane or hyperbolic 3 space may be represented by 2 2 matrices whose entries are rational integers or real or imaginary quadratic integers the theory is extended here to groups operating in h 4 or h 5 and matrices over one of the three basic systems of quaternionic integers quaternionic modular groups are shown to be subgroups of the rotation groups of regular honeycombs of h 4 and h 5 for four dimensional groups the division ring of quaternions is treated as a clifford algebra results in hyperbolic 5 space derive from the homeomorphism of inversive 4 space and the quaternionic projective line

logical structure of fuzzy if then rules abstract this paper provides a logical basis for manipulation with fuzzy if then rules our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to lotfi a zadeh but also other kinds of approximate reasoning methods e g perception based deduction provided that there exists a possibility to characterize them within a formal logical system in contrast with other approaches employing variants of multiple valued first order logic the approach presented here employs fuzzy type theory of v novak which has sufficient expressive power to present the essential concepts and results in a compact elegant and justifiable form within the effectively formalized representation developed here based on a complete logical system it is possible to reconstruct numerous well known properties of cri related fuzzy inference methods albeit not from the analytic point of view as usually presented but as formal derivations of the logical system employed the authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy if then rule bases will be represented formalized and backed up by proof within the well founded logical representation presented here an immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy if then rules are first class citizens of the representation there are clear logically founded definitions for fuzzy if then rule bases to be consistent complete or independent

on some questions concerning the axiomatisation of wnm algebras and their subvarieties in a seminal paper esteva and godo introduced monoidal t norm based logic mtl and some of its prominent extensions such as nm and wnm we notice that nm is axiomatisable from imtl and hence mtl with one variable axioms by instantiating the wnm axiom over one variable this observation leads us here to study the logic axiomatised by extending mtl by this one variable axiom we shall refer to its equivalent algebraic semantics as the variety of ghp algebras for those algebras will be shown to form the largest variety of mtl algebras such that the falsum free reducts of the positive cones of their chains are the most general totally ordered godel hoops among other results we obtain a general description of ghp standard algebras and use the latter to characterise those extensions of wnm that can be obtained from ghp via the same set of extending axioms

rotation invariant t norms where triple rotation and rotation annihilation meet generalizing the notion of a zoom of a t norm not only allows to extend the triple rotation method and its corresponding decomposition but also allows to describe the interface between the triple rotation method and the rotation annihilation method an alternative view on the partition behind the rotation annihilation method is required to make both methods compatible besides zooms we also use contour lines and the companion of a t norm to concisely formulate the results

the logic of tied implications part 1 properties applications and representation a conjunction t ties an implication operator a if the identity a a a b z a t a b z holds a a abdel hamid n n morsi associatively tied implications fuzzy sets and systems 136 2003 291 311 we study the class of tied adjointness algebras which are five connective algebras on two partially ordered sets in which the implications are tied by triangular norms this class contains besides residuated implications several other implications employed in fuzzy logic nevertheless we show that the algebraic inequalities of residuated algebras remain true for our tied implications but in forms that distribute roles over the five connectives of the algebra we apply the properties of tied implications to a generalized modus ponens inference scheme with two successive rules we prove its equivalence to a scheme with one compound rule when both schemata are interpreted by the compositional rule of inference and all connectives are taken from one tied adjointness algebra then we quote another application of this rich theory a notion of many valued rough sets which exhibit the basic mathematical behaviour of the rough sets of pawlak a comparator h is said to be prelinear if it satisfies h y z h z y 1 for all y z hajek we introduce prelinear tied adjointness algebras in which two comparators are prelinear we provide a representation of those algebras as subdirect products of tied adjointness chains on the lines of hajek s representation of bl algebras but our representations are more economical because we employ minimal prime filters on residuated lattices only rather than all prime filters

characterizations of maximal consistent theories in the formal deductive system l nm logic and cantor space a maximal consistent theory is a maximal theory with respect to its consistency the present paper is divided into two parts the first one is devoted to characterize the maximality of a consistent theory in the formal deductive system l which is a logic system equivalent to the nilpotent minimum logic it is proved that each maximal consistent theory in this logic must be the deductive closure of a collection of simple compound formulas hence it follows that there is a one to one correspondence between the set of all maximal consistent theories and the set of evaluations e assigning to each propositional variable p its truth degree e p 0 12 1 the satisfiability theorem and compactness theorem of l are obtained the second part is to investigate the topological structure of the set of all maximal consistent theories over l and the results show that this topological space is a cantor space

topology on bl algebras in this paper by using a special family of filters on a bl algebra l we construct a topology on l show that l is a topological bl algebra find the closure of any subsets of l and obtain some conditions under which that l is a hausdorff space also we prove some theorems about connected components of this topological space and use them to construct a quotient semitopological bl algebras finally we introduce the concept of complete bl algebra and attempt to construct a complete bl algebra from l

fuzzy logic with countable evaluated syntax revisited in this paper we revisit fuzzy logic with evaluated syntax ev l an objection towards it is necessity to introduce logical constants names of truth values for all email protected 0 1 into its language which makes it uncountable hajek has shown in his book metamathematics of fuzzy logic that when seeing ev l as rpl logic where the provability degree is an additional concept then we may confine only to rational logical constants in this paper we show that the same result can be obtained directly in ev l the essential idea is to generalize slightly the concept of a complete fuzzy theory using the technique of ev l

transformations of heun s equation and its integral relations using the transformation theory for the heun equation we find substitutions of variables which preserve the form of the equation for the kernels of integral relations among solutions of the heun equation these transformations lead to new kernels for the heun equation given by single hypergeometric functions lambe ward type kernels and by products of two hypergeometric functions erd lyi type such kernels by a limiting process also afford new kernels for the confluent heun equation

discrete dualities for n potent mtl algebras and 2 potent bl algebras abstract discrete dualities are developed for n potent mtl algebras and for 2 potent bl algebras that is classes of frames or relational systems are defined that serve as dual counterparts to these classes of algebras the frames defined here are extensions of the frames that were developed for mtl algebras in 24 and 25 the additional frame conditions required are given here and also the proofs that discrete dualities hold with respect to such frames the duality also provides an embedding from an n potent mtl algebra or 2 potent bl algebra into the complex algebra of its canonical frame which is a complete algebra in the lattice sense

density elimination density elimination a close relative of cut elimination consists of removing applications of the takeuti titani density rule from derivations in gentzen style hypersequent calculi its most important use is as a crucial step in establishing standard completeness for syntactic presentations of fuzzy logics that is completeness with respect to algebras based on the real unit interval 0 1 this paper introduces the method of density elimination by substitutions for general classes of first order hypersequent calculi it is shown that density elimination by substitutions is guaranteed by known sufficient conditions for cut elimination these results provide the basis for uniform characterizations of calculi complete with respect to densely and linearly ordered algebras standard completeness follows for many first order fuzzy logics using a dedekind macneille style completion and embedding

the ł and ł propositional and predicate logics abstract this paper has two main goals the first goal is to show a different axiomatic system of the and 1 2 propositional logics these propositional logics were introduced in esteva et al arch math logic to appear and they are the combinations of the ukasiewicz and the product logic together with the constant 1 2 in case of the 1 2 logic the second goal is to show an axiomatic system and the completeness theorem of a predicate version of the and 1 2 propositional logics it will be shown that godel product and ukasiewicz predicate logics are contained in the logic

an early approach toward graded identity and graded membership in set theory the paper considers an early approach toward a fuzzy set theory with a graded membership predicate and a graded equality relation which had been developed by the german mathematician klaua in 1965 in the context of the mathematical fuzzy logic mtl of left continuous t norms we discuss some properties of these graded relations we compare the simultaneous recursive definitions of these relations with the very similar approach toward boolean algebra valued interpretations of membership and equality presented in 1967 by scott and r solovay in the context of independence proofs for zf set theory finally we speculate about possible reasons why klaua soon abandoned this approach

prenex normal form in linguistic quantifiers modeled by sugeno integrals in this paper ying s prenex normal form theorem for linguistic quantifiers is strengthened so that it behaves like the one in the classical logic this result further shows elegant logical properties of ying s framework

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

fuzzy sets and formal logics the paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics our focus is on the historical side of these developments

confluent heun equations convergence of solutions in series of coulomb wavefunctions the leaver solutions in series of coulomb wavefunctions for the confluent heun equation are given by two sided infinite series that is by series where the summation index n runs from minus to plus infinity leaver 1986 j math phys 27 1238 first we show that in contrast to the d alembert test under certain conditions the raabe test ensures that the domains of convergence of these solutions include an additional singular point we also consider solutions for a limit of the confluent heun equation for both equations new solutions are generated by transformations of variables finally we discuss the time dependence of the klein gordon equation in two cosmological models and the spatial dependence of the schr dinger equation to a family of quasi exactly solvable potentials for a subfamily of these potentials we obtain infinite series solutions which converge and are bounded for all values of the independent variable in opposition to a common belief

some kinds of c ccq interval valued fuzzy ideals of bci algebras in this paper we first introduce the notions of positive implicative implicative and commutative interval valued fuzzy ideals of bci algebras which are generalizations of positive implicative implicative and commutative fuzzy ideals respectively and investigate some of their related properties the concept of quasi coincidence of an interval valued fuzzy point within an interval valued fuzzy set is introduced in fact the concept is a generalized concept of quasi coincidence of a fuzzy point within a fuzzy set by using this new idea we further introduce the notions of positive implicative implicative and commutative q interval valued fuzzy ideals of bci algebras and investigate some of their related properties some characterization theorems of these generalized interval valued fuzzy ideals are derived the relationship among these generalized interval valued fuzzy ideals of bci algebras is also considered

issues on adjointness in multiple valued logics we contribute to the theory of implications and conjunctions related by adjointness in multiple valued logics we suggest their use in zadeh s compositional rule of inference to interpret generalized modus ponens inference schemata we provide new complete characterizations of implications that distinguish left arguments implications that satisfy the exchange principle divisible conjunctions commutative conjunctions associative conjunctions and triangular norms we also introduce and characterize pseudo strict and pseudo continuous implications and conjunctions and we explore the close relationship between these two notions

an algebraic approach to linguistic hedges in zadeh s fuzzy logic the paper addresses the mathematical modelling of domains of linguistic variables i e term sets of linguistic variables in order to obtain a suitable algebraic structure for the set of truth values of zadeh s fuzzy logic we shall give a unified algebraic approach to the natural structure of domains of linguistic variables which was proposed by ho and wechler fuzzy sets and systems 35 1990 281 and then by ho and nam proc ncst vietnam 9 1 1997 15 logic algebra and computer science vol 46 banach center publications pwn warsaw 1999 p 63 in this approach every linguistic domain can be considered as an algebraic structure called hedge algebra because properties of its unary operations reflect semantic characteristics of linguistic hedges many fundamental properties of refined hedge algebras rh algebras are examined especially it is shown that every rh algebra of a linguistic variable with a chain of the primary terms is a distributive lattice rh algebras with exactly two distinct primary terms one being an antonym of the other will also be investigated and they will be called symmetrical rh algebras it is shown that a class of finite symmetrical rh algebras has a rich enough algebraic structure

continuous weakly cancellative triangular subnorms i their web geometric properties abstract the paper studies web geometric properties of continuous weakly cancellative t subnorms particularly it studies the reidemeister closure condition which is known from web geometry as a visual characterization of those loops that are associative as a result the paper delimits a subset of the domain of a continuous weakly cancellative t subnorm in which the reidemeister closure condition is necessarily satisfied

higher dimensional analogues of the modular and picard groups clifford algebras are used to describe arithmetic groups which are generalizations of the modular and picard groups

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

adjoint negations more than residuated negations negation operators are useful operators which have extensively been studied and used in different fuzzy settings such as in fuzzy logic this paper introduces a new and flexible kind of negation operator based on adjoint triples and which is called adjoint negation relative to a fixed element besides proving several interesting properties the comparison with weak negations studied by trillas esteva et al and pairs of weak negations introduced by georgescu and popescu is presented furthermore adjoint negations are used in order to define the corresponding dual disjunctive to an adjoint conjunctor

parameterizing the semantics of fuzzy attribute implications by systems of isotone galois connections we study the semantics of fuzzy if then rules called fuzzy attribute implications parameterized by systems of isotone galois connections the rules express dependences between fuzzy attributes in object attribute incidence data the proposed parameterizations are general and include as special cases the parameterizations by linguistic hedges used in earlier approaches we formalize the general parameterizations propose bivalent and graded notions of semantic entailment of fuzzy attribute implications show their characterization in terms of least models and complete axiomatization and provide characterization of bases of fuzzy attribute implications derived from data

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

commensurability classes of hyperbolic coxeter groups in this paper we classify all the hyperbolic coxeter n simplex reflection groups up to widecommensurability for all n 3 we also determine all the subgroup relationships among the groups 2002 elsevier science inc all rights reserved ams classification primary 20f55

reasoning about mathematical fuzzy logic and its future this paper is devoted to reasoning about fuzzy logic which is based on various personal observations of the author our goal is to think of the state of the art in mathematical fuzzy logic mfl and to outline some of the tasks on which in the author s opinion mfl should focus in the future in our discussion we will mention not only the basic theory but also its extension called fuzzy logic in broader sense flb the paradigm of the latter is to be the logic of natural human reasoning whose most essential characteristic is the use of natural language besides brief description of flb we will also mention some of its applications on the basis of that we will ponder on other possible directions for research namely the possibility of using fln as a metatheory of fuzzy mathematics as a proper tool for modeling of the main manifestations of the phenomenon of vagueness and as a reasonable tool for developing models of linguistic semantics

term satisfiability in flew algebras fl ew algebras form the algebraic semantics of the full lambek calculus with exchange and weakening we investigate two relations called satisfiability and positive satisfiability between fl ew terms and fl ew algebras for each fl ew algebra the sets of its satisfiable and positively satisfiable terms can be viewed as fragments of its existential theory we identify and investigate the complements as fragments of its universal theory we offer characterizations of those algebras that positively satisfy just those terms that are satisfiable in the two element boolean algebra providing its semantics to classical propositional logic in case of positive satisfiability these algebras are just the nontrivial weakly contractive fl ew algebras in case of satisfiability we give a characterization by means of another property of the algebra the existence of a two element congruence further we argue that positive satisfiability problems in fl ew algebras are computationally hard some previous results in the area of term satisfiability in mv algebras or bl algebras are thus brought to a common footing with known facts on satisfiability in heyting algebras

a characterization of truth functions in the nilpotent minimum logic abstract by introducing a new family of partitions into the n cube 0 1 n the problem of characterizing truth tables of formulas in the nilpotent minimum logic is solved and their normal forms are presented so far only this kind of fuzzy truth functions have normal forms among all fuzzy propositional calculi which are based on left continuous but discontinuous t norm

a graded notion of functionality abstract we will introduce the notion of functionality as a graded property of a fuzzy relation we will extend known results relating to functional fuzzy relations to their graded counterparts and provide new properties w r t various kinds of relational compositions as a contribution to the theory of fuzzy control we will moreover provide an example of use of graded properties in establishing implicative models of fuzzy rule bases where the non graded property fails

spiraling spectra of geodesic lines in negatively curved manifolds given a negatively curved geodesic metric space m we study the asymptotic penetration behaviour of geodesic lines of m in small neighbourhoods of closed geodesics and of other compact convex subsets of m we define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects we prove analogs of the theorems of dirichlet hall and cusick in this context as a consequence we obtain diophantine approximation results of elements of mathbb r mathbb c or the heisenberg group by quadratic irrational ones