EXTENSIONS OF PARTIAL LATTICE-VALUED POSSIBILISTIC MEASURES FROM NESTED DOMAINS

2006 ◽  
Vol 14 (02) ◽  
pp. 175-197 ◽  
Author(s):  
IVAN KRAMOSIL
Keyword(s):  
Complete Lattice ◽  
Possibility Degree ◽  
Partial Lattice ◽  
Set Inclusion ◽  
Power Set ◽  
The Universe ◽  
Nested System

We investigate a partial non-numerical possibilistic measure taking its values in a complete lattice and defined on a nested system of subsets of the universe under consideration. Our aim is to extend this measure conservatively to the power-set of all systems of this universe using the same idea as that when introducing outer measures. Hence, we ascribe to each subset of the universe its minimal (in the sense of set inclusion) covering by a set from the nested domain and define its possibility degree as identical with the value ascribed to this covering. Also analyzed is the case when they are two lattice-valued possibilistic measures on the same universe, each with its own nested domain and our aim is to define one possibilistic measure on the power-set in question in a way sophistically taking profit of the information offered by both the particular partial lattice-valued measures.

scholarly journals On the Weak Order of Coxeter Groups

2019 ◽  
Vol 71 (2) ◽  
pp. 299-336 ◽  
Author(s):  
Matthew Dyer
Keyword(s):  
Complete Lattice ◽  
Coxeter Groups ◽  
Closure Operator ◽  
Root Systems ◽  
Bruhat Order ◽  
Weak Order ◽  
Maximal Subgroups ◽  
Rank Zero ◽  
Power Set ◽  
Rank One

AbstractThis paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of $W$ to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general).

Sets equipollent to their power set in NF

1975 ◽  
Vol 40 (2) ◽  
pp. 149-150 ◽  
Author(s):  
Maurice Boffa
Keyword(s):  
Power Set ◽  
Image Position ◽  
The Universe

In this note we define a class of properties for which the following holds: If we can prove in NF that the property holds for the universe V, then we can prove in NF that it holds for every set equipollent to its power set.Definition. For any stratified formula A and any variable υ which does not occur in A, let Aυ be the formula obtained by replacing in A each quantifier (Qx) by the bounded quantifier (Qx ∈ SCi(υ)), where i is the type of x in A. We will say that a property P(υ) is typed when there is a stratified sentence S such that P(υ) ↔ Sυ holds in NF.Examples of typed properties are: “υ is Dedekind-infinite”, “υ is not well-orderable”. Specker [3] proved that these typed properties hold for the universe V, and C. Ward Henson [1] extended this result to any set equipollent to its power set. We will show that such an extension holds for any typed property.Theorem. For any typed property P(υ):Proof. Fix a bijective map h: υ → SC(υ) and define for i = 0, 1, 2, …, n, … a bijective map hi: υ → SCi(υ) as follows:For every formula A, let A(h) be obtained by replacing in A each atomic part (x ∈ y) by (x ∈ h(y)) and each quantifier (Qx) by (Qx ∈ υ).

Strongly compact cardinals, elementary embeddings and fixed points

1984 ◽  
Vol 49 (3) ◽  
pp. 808-812
Author(s):  
Yoshihiro Abe
Keyword(s):  
Fixed Points ◽  
Elementary Embedding ◽  
Inner Model ◽  
Power Set ◽  
Elementary Embeddings ◽  
Basic Facts ◽  
The Universe ◽  
Normal Ultrafilter ◽  
Strongly Compact Cardinals

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

POWER CLONES AND NON-DETERMINISTIC HYPERSUBSTITUTIONS

2008 ◽  
Vol 01 (02) ◽  
pp. 177-188 ◽  
Author(s):  
K. Denecke ◽  
P. Glubudom ◽  
J. Koppitz
Keyword(s):  
Canonical Extension ◽  
Complex Product ◽  
Original Language ◽  
Tree Transformations ◽  
Power Set ◽  
Tree Transducers ◽  
Tree Languages ◽  
Heterogeneous Algebra ◽  
The Universe ◽  
Tree Transformation

Hypersubstitutions map operation symbols to terms of the corresponding arity. Any hypersubstitution can be extended to a mapping defined on the set Wτ(X) of all terms of type τ. If σ : {fi | i ∈ I} → Wτ(X) is a hypersubstitution and [Formula: see text] its canonical extension, then the set [Formula: see text] is a tree transformation where the original language and the image language are of the same type. Tree transformations of the type Tσ can be produced by tree transducers. Here Tσ is the graph of the function [Formula: see text]. Since the set of all hypersubstitutions of type r forms a semigroup with respect to the multiplication [Formula: see text], semigroup properties influence the properties of tree transformations of the form Tσ. For instance, if σ is idempotent, the relation Tσ is transitive (see [1]). Non-deterministic tree transducers produce tree transformations which are not graphs of some functions. If such tree transformations have the form Tσ, then σ is no longer a function. Therefore, there is some interest to study non-deterministic hypersubstitutions. That means, there are operation symbols which have not only one term of the corresponding arity as image, but a set of such terms. To define the extensions of non-deterministic hypersubstitutions, we have to extent the superposition operations for terms to a superposition defined on sets of terms. Let [Formula: see text] be the power set of the set of all n-ary terms of type τ. Then we define a superposition operation [Formula: see text] and get a heterogeneous algebra [Formula: see text] (ℕ+ is the set of all positive natural numbers), which is called the power clone of type τ. We prove that the algebra [Formula: see text] satisfies the well-known clone axioms (C1), (C2), (C3), where (C1) is the superassociative law (see e.g. [5], [4]). It turns out that the extensions of non-deterministic hypersubstitutions are precisely those endomorphisms of the heterogeneous algebra [Formula: see text] which preserve unions of families of sets. As a consequence, to study tree transformations of the form Tσ, where σ is a non-deterministic hypersubstitution, one can use the structural properties of non-deterministic hypersubstitutions. Sets of terms of type τ are tree languages in the sense of [3] and the operations [Formula: see text] are operations on tree languages. In [3] also another kind of superposition of tree languages is introduced which generalizes the usual complex product of subsets of the universe of a semigroup. We show that the extensions of non-deterministic hypersubstitutions are not endomorphisms with respect to this kind of superposition.

scholarly journals UM SISTEMA DE TABLEAUX ANALÍTICOS PARA A LÓGICA DO PLAUSÍVEL

1969 ◽  
Vol 6 (2) ◽  
Author(s):  
Luiz Henrique Da Cruz Silvestrini
Keyword(s):  
Classical Logic ◽  
Inductive Reasoning ◽  
Deductive System ◽  
Axiomatic System ◽  
Generalized Quantifier ◽  
Power Set ◽  
Number Of Individuals ◽  
The Universe ◽  
Good Number

The Logic of the Plausible was introduced in 1999 by Grácio as a particularization of a family of logical systems characterized by the inclusion of a generalized quantifier in the syntax of the classical logic of predicates, denominated the Modulated Logics. The semantical interpretation of these logics is given by a subset of the power set of the universe. In this particularization of modulated logics, it is included the quantifier of Plausible P that engenders the formalization of a type of inductive reasoning so that "a ‘good’ number of individuals possesses certain property". We introduced a new deductive system for the Logic of the Plausible, denominated TLP, built following the principles of the classical semantical tableaux. Besides, we sketched the equivalence of this new deductive system relative to the axiomatic system originally presented by Grácio.

Set Relations via Families of Scalar Functions and Approximate Solutions in Set Optimization

2020 ◽  
Author(s):  
Giovanni Paolo Crespi ◽  
Andreas H. Hamel ◽  
Matteo Rocca ◽  
Carola Schrage
Keyword(s):  
Complete Lattice ◽  
Optimization Problems ◽  
Multicriteria Decision Making ◽  
Approximate Solutions ◽  
Set Optimization ◽  
Mathematical Economics ◽  
New Approach ◽  
Power Set ◽  
Preordered Set ◽  
Set Relations

Via a family of monotone scalar functions, a preorder on a set is extended to its power set and then used to construct a hull operator and a corresponding complete lattice of sets. Functions mapping into the preordered set are extended to complete lattice-valued ones, and concepts for exact and approximate solutions for corresponding set optimization problems are introduced and existence results are given. Well-posedness for complete lattice-valued problems is introduced and characterized. The new approach is compared with existing ones in vector and set optimization. Its relevance is shown by means of many examples from multicriteria decision making, statistics, and mathematical economics and finance.

An absoluteness principle for Borel sets

1998 ◽  
Vol 63 (2) ◽  
pp. 663-693 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Direct Proof ◽  
General Study ◽  
Closed Set ◽  
Borel Sets ◽  
Borel Hierarchy ◽  
Small Cardinality ◽  
Power Set ◽  
Independence Results ◽  
Notational System ◽  
The Universe

The purpose of these notes is to describe an absoluteness principle due to Jacques Stern and discuss some applications to the general study of Borel sets. This paper will not be engaged in independence results, but in proving outright theorems about the Borel hierarchy.Roughly speaking, Stern's absoluteness principle states that if a certain set can be introduced into the universe by forcing, then it can be introduced by some small forcing notion.The notation , and so on, will be defined in Section 1; this gives a notational system for describing the complexity of Borel sets beyond Fσ or Gδ. The “universe” refers to the totality of all sets. “Forcing” refers to Paul Cohen's technique for, in some sense, changing this totality by the introduction of new sets. Here “small” means relatively small cardinality.The size of this small forcing notion is roughly the ath iteration of the power set operation. Just to get an idea of what this theorem might be saying, we can argue that under certain conditions, if a closed set can be introduced by forcing, then it exists already. There are a number of other qualifications that need to be made to this rough description, and we will come to them later.Unlike, say, Shoenfield absoluteness, Stern's absoluteness can only be made understood in the terminology of forcing. Since forcing is typically associated with the pursuit of independence results, we could easily assume that Stern's work has little relevance in proving positive theorems about the Borel hierarchy.However, this would be untrue. Using abstract and indirect metamathematical arguments, and availing ourselves of Stern's absoluteness principle, we will prove a string of ZFC theorems for which no direct proof is known.

Set theory, philosophy of

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Set Theory ◽  
The Other ◽  
Second Stage ◽  
Other Hand ◽  
Iterative Conception ◽  
Power Set ◽  
After Effect ◽  
Almost All ◽  
The Universe ◽  
Philosophy Of Set Theory

The various attitudes that have been taken to mathematics can be split into two camps according to whether they take mathematical theorems to be true or not. Mathematicians themselves often label the former camp realist and the latter formalist. (Philosophers, on the other hand, use both these labels for more specific positions within the two camps.) Formalists have no special difficulty with set theory as opposed to any other branch of mathematics; for that reason we shall not consider their view further here. For realists, on the other hand, set theory is peculiarly intractable: it is very difficult to give an unproblematic explanation of its subject matter. The reason this difficulty is not of purely local interest is an after effect of logicism. Logicism, in the form in which Frege and Russell tried to implement it, was a two-stage project. The first stage was to embed arithmetic (Frege) or, more ambitiously, the whole of mathematics (Russell) in the theory of sets; the second was to embed this in turn in logic. The hope was that this would palm off all the philosophical problems of mathematics onto logic. The second stage is generally agreed to have failed: set theory is not part of logic. But the first stage succeeded: almost all of mathematics can be embedded in set theory. So the logicist aim of explaining mathematics in terms of logic metamorphoses into one of explaining it in terms of set theory. Various systems of set theory are available, and for most of mathematics the method of embedding is fairly insensitive to the exact system that we choose. The main exceptions to this are category theory, whose embedding is awkward if the theory chosen does not distinguish between sets and proper classes; and the theory of sets of real numbers, where there are a few arguments that depend on very strong axioms of infinity (also known as large cardinal axioms) not present in some of the standard axiomatizations of set theory. All the systems agree that sets are extensional entities, so that they satisfy the axiom of extensionality: ∀x(xЄa ≡ xЄb) → a=b. What differs between the systems is which sets they take to exist. A property F is said to be set-forming if {x:Fx} exists: the issue to be settled is which properties are set-forming and which are not. What the philosophy of set theory has to do is to provide an illuminating explanation for the various cases of existence. The most popular explanation nowadays is the so-called iterative conception of set. This conceives of sets as arranged in a hierarchy of stages (sometimes known as levels). The bottom level is a set whose members are the non-set-theoretic entities (sometimes known as Urelemente) to which the theory is intended to be applicable. (This set is often taken by mathematicians to be empty, thus restricting attention to what are known as pure sets, although this runs the danger of cutting set theory off from its intended application.) Each succeeding level is then obtained by forming the power set of the preceding one. For this conception three questions are salient: Why should there not be any sets other than these? How rich is the power-set operation? How many levels are there? An alternative explanation which was for a time popular among mathematicians is limitation of size. This is the idea that a property is set-forming provided that there are not too many objects satisfying it. How many is too many is open to debate. In order to prevent the system from being contradictory, we need only insist that the universe is too large to form a set, but this is not very informative in itself: we also need to be told how large the universe is.

scholarly journals A New Set Theory for Analysis

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 31
Author(s):  
Juan Ramírez
Keyword(s):  
Real Number ◽  
Number System ◽  
Order Structure ◽  
Natural Numbers ◽  
Real Numbers ◽  
The Real ◽  
Power Set ◽  
Universe Of Sets ◽  
The Universe ◽  
Real Number System

We provide a canonical construction of the natural numbers in the universe of sets. Then, the power set of the natural numbers is given the structure of the real number system. For this, we prove the co-finite topology, C o f ( N ) , is isomorphic to the natural numbers. Then, we prove the power set of integers, 2 Z , contains a subset isomorphic to the non-negative real numbers, with all its defining structures of operations and order. We use these results to give the power set, 2 N , the structure of the real number system. We give simple rules for calculating addition, multiplication, subtraction, division, powers and rational powers of real numbers, and logarithms. Supremum and infimum functions are explicitly constructed, also. Section 6 contains the main results. We propose a new axiomatic basis for analysis, which represents real numbers as sets of natural numbers. We answer Benacerraf’s identification problem by giving a canonical representation of natural numbers, and then real numbers, in the universe of sets. In the last section, we provide a series of graphic representations and physical models of the real number system. We conclude that the system of real numbers is completely defined by the order structure of natural numbers and the operations in the universe of sets.

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