On some chains of fuzzy sets

A partial order relation σ is defined in the set $\text{F}$(X) of the fuzzy sets in X. If this ordering is induced in the subset F(X) of the measurable fuzzy sets in the set X with totally finite positive measure, then fσg implies that the entropy of the fuzyy set f is not less than the entropyof g. By means of this ordering a lattice L on $\text{F}$(X) is defined and a lattice structure is induced in the set of infinite chains in L. Furthermore the set F′(X) of the fuzzy sets of F(X) which assume value in a finite subset of the real interval [0,1] is considered and the following properties are stated: any chain of elements of F′(X) is an infinite sequence of functions convergent in the mean to an integrable function, and the entropy is a valuation of bounded variation on the sublattice of L whose elements are in F′(X). The chains on L can offer a model of a cognitive process in a fuzzy environment when their elements are determined by a sequence of decisions. The limit property traduces the determinism of a such procedure.     

Solution of fuzzy equations with extended operations

Assuming that 1 is any operation defined on a product set X × Y and taking values on a set Z, it can be extended to fuzzy sets by means of Zadeh's extension principle. Given a fuzzy subset C of Z, it is here shown how to solve the equation $\text{A 1 B = C}$ (or $\text{A 1 B ? C}$) when a fuzzy subset A of X (or a fuzzy subset B of Y) is given. The methodology we provide includes, as a special case, the resolution of fuzzy arithmetical operations, i.e. when 1 stands for +, ?, × or ÷, extended to fuzzy numbers (fuzzy subsets of the real line). The paper is illustrated with several examples in fuzzy arithmetic.     

Fuzzy subobjects in a category and the theory of C-sets

In this paper we shall define the concept of a fuzzy subobject of an object in arbitrary categories. This concept is generated by the representation theorem of fuzzy sets. By using fuzzy subobjects one can include most of the fuzzy concepts defined in the literature, such as: fuzzy groups, fuzzy relations and fuzzy convex sets. In the second part of the paper we shall define a new concept; that of a $\text{C}$-set. This concept will generalize that of a fuzzy set and we shall also prove that $\text{C}$-sets can be represented by some sets of functors. More precisely, $\text{C}$-sets form a category which can be represented by a category of functors. The utility of $\text{C}$-sets resides in the fact that one can replace “ordering” by the more general concept of a morphism in category. The new representation of $\text{C}$-sets is weaker than that of fuzzy sets.     

Projections on spaces of invariant functions

Let (X, ∑, μ) be a measure space and S be a semigroup of measure-preserving transformations T:X → X. In case μ(X) < ∞, Aribaud [1] proved the existence of a positive contractive projection P of L¹(μ) such that for every $\text{? ? L}{}^1\text{(μ), Pf}$ belongs to the closure $\text{C}{}_1\text{(?)}\text{in}\text{L}{}^1\text{(μ)}$ of the convex hull $\text{C(?)}$ of the set {$\text{? ○ T:T ? S}$}. In this paper we extend this result in three directions: we consider infinite measure spaces, vector-valued functions, and L^p spaces with 1 ? p < ∞, and prove that P is in fact the conditional expectation with respect to the σ-algebra Λ of sets of ∑ which are invariant with respect to all T?S.     

Eigen fuzzy sets and fuzzy relations

When R is a fuzzy relation between the elements of a finite set X, the fuzzy sets A of X such that R ° A = A (MAX-MIN composition) are called eigen fuzzy sets. The main result of this paper is the determination of the greatest eigen fuzzy set associated with a given fuzzy relation and we give three methods illustrated by an example. We then state that the greatest eigen fuzzy set associated with $\text{R}\text{?}$, the transitive closure of R, is exactly the one associated with R. Finally we describe how to obtain all fuzzy relations keeping invariant a given fuzzy set.     

Fuzzy propositional logics

Fuzzy logic $\text{L}$_∞9 considered in connection with fuzzy sets theory, is a special theory, is a special many valued logic with truth-value sets [0, 1], which has been studied already by Lukasiewicz. We consider also his versions $\text{L}$_m for m ? 2 with finite truth-value sets. In all cases we add two further propositional connectives, one conjunction and one disjunction. For these logics we give a list of tautologies, consider relations between their sets of tautologies, prove their compactness, and mention some further results.     

Fuzzy addition in the L-fuzzy real line

Under the hypothesis L is a chain, we construct a binary operation ⊕ on the L-fuzzy real line $\text{R}$(L) which reduces to the usual addition on $\text{R}$ if ⊕ is restricted to the embedded image of $\text{R}$ in $\text{R}$(L), which yields a partially ordered, abelian cancellation semigroup with identity, and which is jointly fuzzy continuous on $\text{R}$(L). We show ⊕ is unique, i.e. it is the only extension of addition to $\text{R}$(L) which is consistent. We study the relationship between ⊕ and other fuzzy continuous extensions of the usual addition. We also show that fuzzy translation is a weak fuzzy homeomorphism and, under certain conditions, a fuzzy homeomorphism.     

Fuzzy sets as a basis for a theory of possibility

The theory of possibility described in this paper is related to the theory of fuzzy sets by defining the concept of a possibility distribution as a fuzzy restriction which acts as an elastic constraint on the values that may be assigned to a variable. More specifically, if F is a fuzzy subset of a universe of discourse U = {u} which is characterized by its membership function μ_F, then a proposition of the form $\text{“X}\text{is}\text{F”}$, where X is a variable taking values in U, induces a possibility distribution $\text{t}\text{?}{}_{\mathrm{x}}$ which equates the possibility of X taking the value u to μ_F(u)—the compatibility of u with F. In this way, X becomes a fuzzy variable which is associated with the possibility distribution $\text{t}\text{?}{}_{\mathrm{x}}$ in much the same way as a random variable is associated with a probability distribution. In general, a variable may be associated both with a possibility distribution and a probability distribution, with the weak connection between the two expressed as the possibility/probability consistency principle.A thesis advanced in this paper is that the imprecision that is intrinsic in natural languages is, in the main, possibilistic rather than probabilistic in nature. Thus, by employing the concept of a possibility distribution, a proposition, p, in a natural language may be translated into a procedure which computes the probability distribution of a set of attributes which are implied by p. Several types of conditional translation rules are discussed and, in particular, a translation rule for propositions of the form $\text{“X}\text{is}\text{F}$ is α-possible”, where α is a number in the interval [0,1], is formulated and illustrated by examples.     

Existence and uniqueness of statistical measures for solution processes for linear-stochastic differential equations

Structure is developed on the set of real-valued stochastic processes in terms of the authors recently defined statistical measures making explicit an $\text{L}{}_{\mathrm{p}}{}^{\mathrm{n}}\text{(Ω, T)}$-calculus over the structure. This proves that the stochastic-differential equation $\text{L}$y=x, where x is a stochastic process and $\text{L}$ is an nth order linear-stochastic differential operator with up to n ? 1 stochastic-process coefficients, is solved by Adomian's series, and finally, establishes the existence and uniqueness of the statistical measures of the solution process.     

Completion of L-fuzzy relations

It is shown that the category $\text{L}\text{?}$ of complete L-similarities on L-sets is a full reflective subcategory of $\text{R}$(L) (L-fuzzy graphs); $\text{L}\text{?}$ is equivalent to $\text{Lh}$ (L)(sheaves on L). Connections with other known “fuzzy” categories are also studied.     

A new fuzzy compactness defined by fuzzy nets

Let (Ω, $\text{A}$, μ) be a probability space and let $\text{B}$ be a subsigma algebra of $\text{A}$. Let A= L_∞Ω, $\text{A}$, μ , let A= L_∞Ω, $\text{B}$, μ, and let f?A. It is shown that best L_∞-approximations of f by elements of B comprise an interval in B; that is, there exists $\text{f}\text{,}\text{f}\text{?B}$ such that a function g?B is a best L_∞-approximation to f if and only if $\text{f}\text{? g ?}\text{f}$ a.e. on Ω. The difference, $\text{f}\text{? f}$, of $\text{f}$ and f is completely characterized in terms of special sets that have been developed in [2]. Then it is established that the best best L_∞-approximation, f_{$\text{B}$,∞}, to f by elements of B is the average of $\text{f}$ and $\text{f}$, where the function f_{$\text{B}$,∞} is defined by f_{$\text{B}$,∞}(ω) lim_{p → ∞}f_$\text{B}$,P(ξ) and f_$\text{B}$,P denotes the best L_p-approximation to f elements of L_p(Ω, $\text{B}$, μ).     

Measures of fuzzy sets and measures of fuzziness

The measures presented in this paper are defined by using Weber's concept of decomposable measures m of crisp sets, having in particular the Archimedean decomposable operations in view (Section 2). Measures $\text{m}$ of fuzzy sets are introduced as integrals with respect to m. For the Archimedean cases, Weber's integral will be used as alternative to Sugeno's and Choquet's concepts (Section 3). What ‘fuzziness’ means will be described by functions of fuzziness F (another name: entropy N-functions) with respect to a negation. In addition to the types of functions of fuzziness which are induced by concave functions, we discuss also the ones which are induced by fuzzy connectives (Section 4). Now, using m for measuring the ‘importance of items’ and F for the ‘fuzziness’ of the possible values of a fuzzy set ?, m?(F ° ?) serves us as a measure of the fuzziness $\text{F}$? of ?. The concepts of De Luca and Termini, Capocelli and De Luca, Kaufmann, Knopfmacher, Loo, Gottwald, Dombi and, under the restriction to the Archimedean cases, also the concepts of Trillas and Riera and Yager turn out to be special cases (Section 5).     

Principle of inclusion-exclusion on semilattices

We introduce an enumeration theorem under lattice action. Let L be a finite semilattice and Ω be a nonempty set. Let $\text{f: L → P(Ω)}$ be a map satisfying f(x ? y) ? f(x) ∩ f(y), where ? and $\text{P}$(Ω) mean “join” and the power set of Ω, respectively. Then

$\text{m}\text{∪}\text{x?L}\text{?(x)}\text{=}\text{Σ}\text{c?C}\text{(?1)}{}^{\mathrm{l(c)}}\text{m}\text{∩}\text{x?c}\text{?(x)}$

, where C is the set of all chains in L and l(c) denotes the length of a chain c. Also the theorem can be dualized. Furthermore, we describe two applications of the theorem to a Boolean lattice of sets and a partition lattice of a set.     

Difference operators and extended truth vectors for discrete functions

Discrete functions are mappings ? of a finite set $\text{d}\text{}$ into a lattice $\text{L}$. Prime blocks and prime antiblocks generalize for discrete functions the well known concepts of prime implicants and of prime implicates for Boolean functions. A lattice difference operator is defined for discrete functions which, together with the concept of extended vector, allows us to derive new attractive algorithms for obtaining the prime blocks and antiblocks of a discrete function. Applications of the theory to p-symmetric Boolean functions and to transient analysis of binary switching networks are mentioned.     

When is an operator the integral of a given spectral measure?

For a closed densely defined operator T on a complex Hilbert space $\text{H}$ and a spectral measure E for $\text{H}$ of countable multiplicity q defined on a σ-algebra $\text{B}$ over an arbitrary space Λ we give three conceptually differing but equivalent answers to the question asked in the title of the paper (Theorem 1.5). We then study the simplifications which accrue when T is continuous or when q = 1 (Sect. 4). With the aid of these results we obtain necessary and sufficient conditions for T to be the integral of the spectral measure of a given group of unitary operators parametrized over a locally compact abelian group Γ (Sect. 5). Applying this result to the Hilbert space $\text{H}$ of functions which are L₂ with respect to Haar measure for Γ, we derive a generalization of Bochner's theorem on multiplication operators (Sect. 6). Some results on the multiplicity of indicator spectral measures over Γ are also obtained. When Γ = $\text{R}$ we easily deduce the classical theorem about the commutant of the associated self-adjoint operator (Sect. 7).     

Fuzzy Boolean algebras

The purpose of this paper is to generalize the following situation: from the concrete structure $\text{B}$, we define the notion of Boolean algebras; the Stone representation theorem allows us to replace the algebraic study of Boolean algebras by a topological one. Let E be a non-empty set, and J a non-empty ordered set. Note $\text{B}$ the set of all fuzzy subsets of (E,J). We shall introduce the concept of fuzzy Boolean algebra and find a representation theorem. But it will be difficult to speak of the dual fuzzy topological space of a fuzzy Boolean algebra as we shall see further, except in certain particular cases.     

Green's function,Jensen measures,and bounded point evaluations

A compactly supported measure μ on the complex plane C is called a Jensen measure for 0 if $\text{log}\text{¦P(0)¦ ? ∝}\text{log}\text{¦P(z)¦dμ(z)}$ for every polynomial P. H²(μ) denotes the closure of the polynomials in L²(μ). We obtain the result that if μ is not the point mass at 0, then the functions in H²(μ) are analytic on an open set which contains 0 and whose closure contains the support of μ. The primary tool used to obtain this result is a generalized Green's function for a measure, and we also derive some of its properties.     

On the irreducibility of a class of polynomials,III

This work is a continuation and extension of our earlier articles on irreducible polynomials. We investigate the irreducibility of polynomials of the form g(f(x)) over an arbitrary but fixed totally real algebraic number field $\text{L}$, where g(x) and f(x) are monic polynomials with integer coefficients in $\text{L}$, g is irreducible over $\text{L}$ and its splitting field is a totally imaginary quadratic extension of a totally real number field. A consequence of our main result is as follows. If g is fixed then, apart from certain exceptions f of bounded degree, g(f(x)) is irreducible over $\text{L}$ for all f having distinct roots in a given totally real number field.     

The inverse balayage problem for Markov chains,part II

This paper continues the study of the inverse balayage problem for Markov chains. Let X be a Markov chain with state space $\text{A ? B}{}_2$, let v be a probability measure on B₂ and let M(v) consist of probability measures μ on A whose X-balayage onto B₂ is v. The faces of the compact, convex set M(v) are characterized. For fixed μ?M(v) the set M(μ,v) of the measures ? of the form $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S) ? ·}}$, where S is a randomized stopping time, is analyzed in detail. In particular, its extreme points and edge are explicitly identified. A naturally defined reversed chain X, for which v is an inverse balayage of μ, is introduced and the relation between X and X^ is studied. The question of which $\text{? ? M(μ, v)}$ admit a natural stopping time $\text{S}{}_{\mathrm{?}}$ of X (not involving an independent randomization) such that $\text{?(·) = P}{}^{\mathrm{\mu }}\text{{X(S}{}_{\mathrm{?}}\text{) ? ·}}$, is shown to have rather different answers in discrete and continuous time. Illustrative examples are presented.