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In fuzzy set theory, the aggregation is the process that combines input fuzzy sets into a single output fuzzy set. In this manner, an aggregation operator is an important tool in the fuzzy set theory and its applications. The purpose of this study is to present some algebraic operators among neutrosophic enthalpy values and to provide some aggregation operators with the help of general t-norms and t-conorms which produce a new theoretical base in the fuzzy environment. An enthalpy value is the information energy expressed by the complement of the Shannon’s entropy and a neutrosophic enthalpy set is characterized with a truth, an indeterminacy and a falsity function defined on a universal set to $$[0,1]^{2}$$ . The first component of each function is the average of the truth, the indeterminacy and the falsity sequence of a neutrosophic multi-valued set, respectively, and the second component of each function is the fuzzy complement of the normalized Shannon’s entropy of the truth, the indeterminacy and the falsity of the same neutrosophic multi-valued set, respectively. Therefore, a neutrosophic enthalpy set contains both the level of the mean of the data and the degree of uncertainty of the data via enthalpy. Then, by using Algebraic and Einstein t-norms and t-conorms we give a multi-criteria decision making method based on these aggregation operators and a score function. This method is applied to a multi-criteria decision making problem with neutrosophic enthalpy set information and the comparison analysis is given with the existing methods to show the efficiency and sensitivity of the proposed method.

In economic science, a new scientific direction has appeared, associated with the so-called prospects theory, which has been among the leading for decision-making under conditions of uncertainty, not only for economic science. The fuzzy set theory, in which the fuzzy set (FS) is one of the types of numerous Non-factors, needs a qualitative modernization since we have a limited possibility to solve many classes of problems. The usage of fuzzy sets is almost universal for solving problems in different fields if there is uncertainty, inaccuracy. At the same time, the difficulty in choosing the membership function should be considered. The process of choosing an MF by a person (expert) is based on common sense, experience, and is not always rational. For solving tasks in conditions of uncertainty, the authors proposed the use of the tensor methodology. The authors showed the possibility to form a subset of ordered pairs based on the tensorization of the interval of a universal set with subsequent decomposition. It was suggested to use Toeplitz matrix, most effectively modeling fuzziness, as one of the ways of tensorization and taking into account the phenomenon of fuzziness. The universal set (US) on which the FS is formed contains, in the tensor format, hidden information that can be used to make a decision no less effectively than the heuristically assigned membership function, in addition, the existence of a formally calculated subset of ordered pairs (SsOP) can serve as a convincing comparative example: Having this SsOP, you can objectively refuse to assign a heuristic MF; SsOP has significantly less interval uncertainty. Alternative SsOP are formed in a uniform manner. Basic characteristics for FS and SsOP comparison are quite close or practically coincide. Examples that illustrate the higher efficiency of SsOP usage in comparison with standardly spaced FS are given.

The original model of rough sets was advanced by Pawlak, which was mainly involved with the approximation of things using an equivalence relation on the universal set of his approximation space. In this paper, two kinds of approximation operators via ideals which represent extensions of Pawlak’s approximation operator have been presented. In both kinds, the definitions of upper and lower approximations based on ideals have been given. Moreover, a new type of approximation spaces via two ideals which is called bi-ideal approximation spaces was introduced for the first time. This type of approximations was analyzed by two different methods, their properties are investigated, and the relationship between these methods is proposed. The importance of these methods was its dependent on ideals which were topological tools, and the two ideals represent two opinions instead of one opinion. At the end of the paper, an applied example had been introduced in the chemistry field by applying the current methods to illustrate the definitions in a friendly way.

Problem solving under uncertainty in the form of fuzzy mathematics based on methods and models of fuzzy set theory with limited possibilities for determining (assigning) membership functions is considered. A method for solving problems of this type is proposed. The method is to determine hidden knowledge in the form of subsets of ordered pairs calculated using a singular value decomposition of special (Toeplitz, Hankel, etc.) matrices formed based on the universal set. Examples that illustrate the effectiveness of the proposed method are given.

We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols to characterize the QECC performance measured in terms of the worst-case entanglement fidelity. The theorem is applicable to a large class of decoherence models, including independent erasure and depolarizing noise. Our approach is unorthodox, as instead of following the established path of utilizing QECCs to mitigate noise in quantum metrological protocols, we apply methods of quantum metrology to explore the limitations of QECCs., 4+5 pages, 2 figures

The Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the algebraic properties of extensions of this system by introducing the notion of the extended Shi–Aharonov structure. The quotient of this structure is isomorphic to a structure based on a particular set of real numbers (the Bloch hypersphere). The aim of this paper is to bring together researchers working in different areas (quantum circuits, algebra, geometry) opening up new perspectives. The algebraic approach may be useful in circuit analysis.

This paper focuses on the labeled fuzzy approximation space, which is considered as a relational structure consisting of a nonempty universal set and some fuzzy relations. To deduce knowledge hidden in the labeled fuzzy approximation space, based on the notion of bisimulations, lower and upper fuzzy rough approximation operators are constructed. Then basic properties of the fuzzy rough approximation operators are investigated. When the largest bisimulation is a trivial identity relation in some cases, the concept of simulations is proposed. Moreover, the lower and upper fuzzy rough relation approximation operators are first proposed and properties of the new operators are examined. Finally, the relationships between two kinds of approximations are discussed.

Discuss the real line fuzzy concept into multidimensional based on the reference function so as to get new imprecise numbers called the two-dimensional and three-dimensional imprecise numbers and their complements. Two and three dimensional imprecise numbers are obtained in the form of Cartesian product of fuzzy numbers. To study their character some necessary definitions like partial presence, construction of membership function, membership value ,Indicator function etc. of two and three-dimensional imprecise numbers are defined with own notation. As per as possible, try to show all the properties of classical set theory that can be hold good in the present imprecise numbers with some examples. Set Operations are defined by maximum and minimum operators just like defined in the real line imprecise numbers. Further bring out a few graphical examples to verify the intersection and union of two and three dimensional imprecise numbers are the empty and the universal set respectively. Basically Intersection and union are the operators to obtain their properties.

The rough set philosophy is based on the concept that there is some information associated with each object of the universe. The set of all objects of the universe under consideration for particular discussion is considered as a universal set. So, there is a need to classify objects of the universe based on the indiscernibility relation (equivalence relation) among them. In the view of granular computing, rough set model is researched by single granulation. The granulation in general is carried out based on the equivalence relation defined over a universal set. It has been extended to multi-granular rough set model in which the set approximations are defined by using multiple equivalence relations on the universe simultaneously. But, in many real life scenarios, an information system establishes the relation with different universes. This gave the extension of multi-granulation rough set on single universal set to multi-granulation rough set on two universal sets. In this paper, we define multi-granulation rough set for two universal sets U and V. We study the algebraic properties that are interesting in the theory of multi-granular rough sets. This helps in describing and solving real life problems more accurately.

In recent years, homomorphisms have been exploited to compare the structures and properties of two generalized information systems. Some of these homomorphisms are based on consistent functions, which are a class of special mappings between universal sets. The purpose of this paper is to unify and extend the consistent functions in the literature into the framework of neighborhood systems. After introducing the notion of consistent functions with respect to neighborhood systems, we explore some important properties of the extended consistent functions such as preserving the inverse images and the intersections of neighborhoods. Our results provide a sound basis for further investigating neighborhood systems via homomorphisms.

The notion of rough set captures indiscernibility of elements in a set. But, in many real life situations, an information system establishes the relation between different universes. This gave the extension of rough set on single universal set to rough set on two universal sets. In this paper, we introduce rough equality of sets on two universal sets and rough inclusion of sets employing the notion of the lower and upper approximation. Also, we establish some basic properties that refer to our knowledge about the universes.

This article studies three most basic systems of truth as well as their subsystems (and one extra system) over set theory ZF possibly with AC or the axiom of global choice GC , and then correlates them with subsystems of Morse–Kelley class theory MK . The article aims at making an initial step towards the axiomatic study of truth in set theory in connection with class theory. Some new results on the side of class theory, such as conservativity, forcing and some forms of the reflection principle, are also presented. The equivalence results among systems of truth and classes obtained in this article are summarized in Theorem 104 (Section 7.1), Theorem 107 (Section 7.2) and Theorem 108 (Section 8).

In this paper, the concept of intension is used to introduce two types of ordering relations based on information that generates a cover for the universal set. These types of ordering relations are distinct from the well-known ordering relation based on set inclusion. For these ordering relations, we consider the algebraic structures that arise in various types of covers. We show that in the case of a representative cover, the algebraic structure resulting from the lower intension inclusion is a double Stone algebra, while in the case of a reduced cover, it is a Boolean algebra. In addition, the algebraic structure resulting from the upper intension inclusion in the case of a representative cover is a Boolean algebra, and in the case of a reduced cover, the two Boolean algebraic structures from lower and upper intension inclusions are isomorphic.

In this paper we summarize some results about sets in Frege structures. The resulting set theory is discussed with respect to its historical and philosophical significance. This includes the treatment of diagonalization in the presence of a universal set.

The asymptotic behavior of the Shannon’s function LB(n) is studied for complexity of n-variable predicate implementation with the use of predicate circuits over arbitrary complete basis B. A new definition of the reduced weight of the predicate is introduced, regarding it as a solution of a specific linear programming problem based on a system of predicate’s generalized variables. New, more exact upper elements for LB(n) in a number of bases are acquired by the means of special decompositions of initial predicates using universal sets of predicates constructed for circuits consisting of bases elements with minimal reduced weight.

For two universal sets U and V, we define the concept of solitary set for any binary relation from U to V. Through the solitary sets, we study the further properties that are interesting and valuable in the theory of rough sets. As an application of crisp rough set models in two universal sets, we find solutions of the simultaneous Boolean equations by means of rough set methods. We also study the connection between rough set theory and Dempster-Shafer theory of evidence. In particular, we extend some results to arbitrary binary relations on two universal sets, not just serial binary relations. We consider the similar problems in fuzzy environment and give an example of application of fuzzy rough sets in multiple criteria decision making in the case of clothes.

We study the extension OST ( E , P ) of Feferman’s operational set theory OST provided by adding operational versions of unbounded existential quantification and power set and determine its proof-theoretic strength in terms of a suitable theory of sets and classes.

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.

In the paper, we discuss several hitherto open problems concerning classes in nonstandard set theory.

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. It is shown that these methods are very useful for the problem of constructing a universal set of gates for quantum computations. The well-known result that the set of all one-bit gates, together with no more than one two-bit gate, is universal is considered from the control theory viewpoint.

This paper deals with ordinal numbers in an algebra-valued model of a paraconsistent set theory. It is proved that the collection of all ordinals is not a set in this model which is dissimilar to the other existing paraconsistent set theories. For each ordinal α of classical set theory α-like elements are defined in the mentioned algebra-valued model whose collection is not singleton. It is shown that two α-like elements (for same α) may perform conversely to validate a given formula of the corresponding paraconsistent set theory.

In this chapter, we introduce the concept of intuitionistic fuzzy parameterised intuitionistic fuzzy soft (ifpifs) sets and their operations with examples. We also define the approximate functions of ifpifs-set from the intuitionistic fuzzy parameterised set to the intuitionistic fuzzy subsets [1] of universal set. Lastly, we construct an ifpifs-set decision-making problem and try to solve the problem.

Theory and Decision, 60 (2-3), ISSN:0040-5833, ISSN:1573-7187

In [7], a naive set theory is introduced based on a polynomial time logical system, Light Linear Logic (LLL). Although it is reasonably claimed that the set theory inherits the intrinsically polytime character from the underlying logic LLL, the discussion there is largely informal, and a formal justification of the claim is not provided sufficiently. Moreover, the syntax is quite complicated in that it is based on a non-traditional hybrid sequent calculus which is required for formulating LLL.

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all.

After introducing the large set notion of Mahloness, this paper shows that constructive set theory with an axiom asserting the existence of a Mahlo set has a realizability interpretation in an extension of Martin-Lof type theory developed by A. Setzer.

A structural (as opposed to Zadeh’s quantitative) approach to fuzziness is given, based on the operator “very”, which is added to the language of set theory together with some elementary axioms about it. Due to the axiom of foundation and to a lifting axiom, the operator is proved trivial on the cumulative hierarchy of ZF. So we have to drop either foundation or lifting. Since fuzziness concerns complemented predicates rather than sets, a class theory is needed for the very operator. And of them the Kelley-Morse (KM) theory is more appropriate for reasons of class existence. Several definable realizations of the very-operator are presented in KM−. In the last section we consider the operator “very” without the lifting axiom on classes of urelements. To each structurally fuzzy set X a traditional quantitative fuzzy set X is assigned – its quantitative representation. This way we are able partly to recover ordinary fuzzy sets from the structurally fuzzy ones.

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory (hence acceptable from a constructive and generalized-predicative standpoint). In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned.

We use covers of the universal set to define approximation operators on the power set of the given set. In Section 1, we determine basic properties of the upper approximation operator and show how it can be used to give algebraic structural properties of certain subsets. We define a particular cover on the set of ideals of a commutative ring with identity in such a way that both the concepts of the (fuzzy) prime spectrum of a ring and rough set theory can simultaneously be brought to bear on the study of (fuzzy) ideals of a ring.

A theorem of Enayat's concerning models of ZFC which had been proved using several different additional set-theoretical hypotheses is shown here to be absolute.

The underlying set theory of the Z-specification language is said to be a typed set theory. There are two interpretations of this to be found in the literature. One is based upon Simple Type theory and the other on set theory plus a type assignment system. We establish their equivalance.

A set theory called Real Set Theory is defined in which Generalized Continuum Hypothesis and Axiom of Choice hold good.

We explore a non-classical, universal set theory, based on a purely ``structural'''' conception of sets. A set is a transfinite process of unfolding of an arbitrary (possibly large) binary structure, with identity of sets given by the observational equivalence between such processes. We formalize these notions using infinitary modal logic, which provides partial descriptions for set structures up to observational equivalence. We describe the comprehension and topological properties of the resulting set-theory, and we use it to give non-classical solutions to classical paradoxes, to prove fixed-point theorems that relate recursion and corecursion, to formalize ``super-large'''', reflexive categories and ``super-large'''' circular models, and to provide ``natural'''' solutions for domain equations.

We shall develop, in this paper, the hybrid set theory extending some notions of the hybrid real number system considered in [V. Lakshmikantham, D. Kovach, Hybrid real number system, Nonlinear Analysis: Hybrid Systems (2006), doi:10.1016/nahs.2006.07.001 ], so as to shed some light on the existing set theory and strengthen it. We shall also list some ideas of set theory that existed long before Cantor’s discovery [G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (P.E.B., Trans.), The Open Court Publishing Co., la Salle, Illinois, 1952].

The purpose of this paper is to provide a simple type-free set theory which can be used to give the various readings of collective/distributive sentences.

Within a categorical framework for primitive recursion, equality between p.r. maps is shown to be definable by suitable p.r. equality predicates. Equivalence is shown between a direct categorical formalization of classical p.r. functions and p.r. maps in the sense of Lawvere and Freyd. An extension of the theory is shown to admit a ‘universal set’ containing all objects of the extended theory of subobjects.

One of the main goals of this paper is to give a construction of realizability models for predicative constructive set theories in a predicative metatheory. We will use the methods of algebraic set theory, in particular the results on exact completion from van den Berg and Moerdijk (2008) [5]. Thus, the principal results of our paper are concerned with the construction of an extension of a category with small maps by a category of assemblies, again equipped with a class of maps, and to show that this extension construction preserves those axioms for a class of maps necessary to produce models of the relevant set theories in the exact completion of this category of assemblies.

We share with Foulis and Randall the evangel that it is not orthomodular posets or the like, but manuals of operations that are of primary importance in the foundations of the empirical sciences. In sharp contrast to them, we regard an operation not as a set of possible outcomes, but as a complete Boolean algebra of observable events, which we adopt, following the lines of Davis and of Takeuti, as a building block of our empirical set theory. Just as a smooth manifold is covered by open subsets of a Euclidean space interconnected by smooth mappings, our empirical set theory is covered by the Scott-Solovay universesV(B) over complete Boolean algebrasB interconnected by geometric morphisms. Using the nomenclature of topos theory, our empirical set theory is a subcategory of the categoryBIop of Boolean localic toposes and geometric morphisms. It is shown that in this set theory observables can be identified with real numbers. This is the first step of formal development of Davis' ambitious program.