Field Of Sets
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Field Of Sets
In mathematics, a field of sets is a mathematical structure consisting of a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X called an algebra over X that contains the empty set as an element, and is closed under the operations of taking complements in X, finite unions, and finite intersections. Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over X" is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory. Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets. Definitions A field of sets is a pair ( X, \mathcal ) consisting of a set X and a family \mathcal of subsets of X, called an algebra over X, that has the following properties: : X \setminus F \in \mathcal for all F \in \mathcal. as an element: \varnot ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ...
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σ-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections. The pair (''X'', Σ) is called a measurable space. A σ-algebra is a type of set algebra. An algebra of sets needs only to be closed under the union or intersection of ''finitely'' many subsets, which is a weaker condition. The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ- ...
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Dedekind Cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two sets ''A'' and ''B'', such that all elements of ''A'' are less than all elements of ''B'', and ''A'' contains no greatest element. The set ''B'' may or may not have a smallest element among the rationals. If ''B'' has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between ''A'' and ''B''. In other words, ''A'' contains every rational number less than the cut, and ''B'' contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rat ...
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Filter (set Theory)
In mathematics, a filter on a set X is a family \mathcal of subsets such that: # X \in \mathcal and \emptyset \notin \mathcal # if A\in \mathcal and B \in \mathcal, then A\cap B\in \mathcal # If A,B\subset X,A\in \mathcal, and A\subset B, then B\in \mathcal A filter on a set may be thought of as representing a "collection of large subsets". Filters appear in order, model theory, set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book '' Topologie Générale'' as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the ...
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Ideal (order Theory)
In mathematical order theory, an ideal is a special subset of a partially ordered set (poset). Although this term historically was derived from the notion of a ring ideal of abstract algebra, it has subsequently been generalized to a different notion. Ideals are of great importance for many constructions in order and lattice theory. Basic definitions A subset of a partially ordered set (P, \leq) is an ideal, if the following conditions hold: # is non-empty, # for every ''x'' in and ''y'' in ''P'', implies that ''y'' is in  ( is a lower set), # for every ''x'', ''y'' in , there is some element ''z'' in , such that and  ( is a directed set). While this is the most general way to define an ideal for arbitrary posets, it was originally defined for lattices only. In this case, the following equivalent definition can be given: a subset of a lattice (P, \leq) is an ideal if and only if it is a lower set that is closed under finite joins ( suprema); that is, it is non ...
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Order Theory
Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and provides basic definitions. A list of order-theoretic terms can be found in the order theory glossary. Background and motivation Orders are everywhere in mathematics and related fields like computer science. The first order often discussed in primary school is the standard order on the natural numbers e.g. "2 is less than 3", "10 is greater than 5", or "Does Tom have fewer cookies than Sally?". This intuitive concept can be extended to orders on other sets of numbers, such as the integers and the reals. The idea of being greater than or less than another number is one of the basic intuitions of number systems (compare with numeral systems) in general (although one usually is also interested in the actual differenc ...
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Stone's Representation Theorem For Boolean Algebras
In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space. Stone spaces Each Boolean algebra ''B'' has an associated topological space, denoted here ''S''(''B''), called its Stone space. The points in ''S''(''B'') are the ultrafilters on ''B'', or equivalently the homomorphisms from ''B'' to the two-element Boolean algebra. The topology on ''S''(''B'') is generated by a (closed) basis consisting of all sets of the form \, where ''b'' is an element of ''B''. This is the topology of pointwise convergence of nets of homomorphisms into the two-element Boolean algebra. For every Boolean algebra ''B'', ''S''(''B'') is a compact totally ...
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Ultrafilter (set Theory)
In the mathematical field of set theory, an ultrafilter is a ''maximal proper filter'': it is a filter U on a given non-empty set X which is a certain type of non-empty family of subsets of X, that is not equal to the power set \wp(X) of X (such filters are called ) and that is also "maximal" in that there does not exist any other proper filter on X that contains it as a proper subset. Said differently, a proper filter U is called an ultrafilter if there exists proper filter that contains it as a subset, that proper filter (necessarily) being U itself. More formally, an ultrafilter U on X is a proper filter that is also a maximal filter on X with respect to set inclusion, meaning that there does not exist any proper filter on X that contains U as a proper subset. Ultrafilters on sets are an important special instance of ultrafilters on partially ordered sets, where the partially ordered set consists of the power set \wp(X) and the partial order is subset inclusion \,\subs ...
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Complete Boolean Algebra
In mathematics, a complete Boolean algebra is a Boolean algebra in which every subset has a supremum (least upper bound). Complete Boolean algebras are used to construct Boolean-valued models of set theory in the theory of forcing. Every Boolean algebra ''A'' has an essentially unique completion, which is a complete Boolean algebra containing ''A'' such that every element is the supremum of some subset of ''A''. As a partially ordered set, this completion of ''A'' is the Dedekind–MacNeille completion. More generally, if κ is a cardinal then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum. Examples Complete Boolean algebras *Every finite Boolean algebra is complete. *The algebra of subsets of a given set is a complete Boolean algebra. *The regular open sets of any topological space form a complete Boolean algebra. This example is of particular importance because every forcing poset can be considered as a topological s ...
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Atomic (order Theory)
In the mathematical field of order theory, an element ''a'' of a partially ordered set with least element 0 is an atom if 0 < ''a'' and there is no ''x'' such that 0 < ''x'' < ''a''. Equivalently, one may define an atom to be an element that is minimal among the non-zero elements, or alternatively an element that covers the least element 0.

# Atomic orderings

Let <: denote the in a partially ordered set. A partially ordered set with a least element 0 is atomic if every element ''b'' > 0 has an atom ''a'' below it, that is, there is some ''a'' such that ''b'' ≥ ''a'' :> ''0''. Every finite parti ...
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Quotient Set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Exa ...