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Partially Ordered Ring
In abstract algebra, a partially ordered ring is a ring (''A'', +, ·), together with a ''compatible partial order'', that is, a partial order \,\leq\, on the underlying set ''A'' that is compatible with the ring operations in the sense that it satisfies: x \leq y \text x + z \leq y + z and 0 \leq x \text 0 \leq y \text 0 \leq x \cdot y for all x, y, z\in A. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring (A, \leq) where partially ordered additive group is Archimedean. An ordered ring, also called a totally ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a total order. An l-ring, or lattice-ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a lattice order. Properties The additive group of a partially ordered ring is always a partially ordered group. The set of non-negative elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Commutative Ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Commutative rings appear in the following chain of subclass (set theory), class inclusions: Definition and first examples Definition A ''ring'' is a Set (mathematics), set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Isabelle (theorem Prover)
The Isabelle automated theorem prover is a higher-order logic (HOL) theorem prover, written in Standard ML and Scala. As a Logic for Computable Functions (LCF) style theorem prover, it is based on a small logical core (kernel) to increase the trustworthiness of proofs without requiring, yet supporting, explicit proof objects. Isabelle is available inside a flexible system framework allowing for logically safe extensions, which comprise both theories and implementations for code-generating, documenting, and specific support for a variety of formal methods. It can be seen as an integrated development environment (IDE) for formal methods. In recent years, a substantial number of theories and system extensions have been collected in the Isabelle ''Archive of Formal Proofs'' (Isabelle AFP). Isabelle was named by Lawrence Paulson after Gérard Huet's daughter. The Isabelle theorem prover is free software, released under the revised BSD license. Features Isabelle is generic: it prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Library (computing)
In computing, a library is a collection of System resource, resources that can be leveraged during software development to implement a computer program. Commonly, a library consists of executable code such as compiled function (computer science), functions and Class (computer programming), classes, or a library can be a collection of source code. A resource library may contain data such as images and Text string, text. A library can be used by multiple, independent consumers (programs and other libraries). This differs from resources defined in a program which can usually only be used by that program. When a consumer uses a library resource, it gains the value of the library without having to implement it itself. Libraries encourage software reuse in a Modular programming, modular fashion. Libraries can use other libraries resulting in a hierarchy of libraries in a program. When writing code that uses a library, a programmer only needs to know how to use it not its internal d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Axiom Of Choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from each set, even if the collection is infinite. Formally, it states that for every indexed family (S_i)_ of nonempty sets (S_i as a nonempty set indexed with i), there exists an indexed set (x_i)_ such that x_i \in S_i for every i \in I. The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem. The axiom of choice is equivalent to the statement that every partition has a transversal. In many cases, a set created by choosing elements can be made without invoking the axiom of choice, particularly if the number of sets from which to choose the elements is finite, or if a canonical rule on how to choose the elements is available — some distinguishing property that happens to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Category (mathematics)
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. ''Category theory'' is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Image (mathematics)
In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each Element (mathematics), element of a given subset A of its Domain of a function, domain X produces a set, called the "image of A under (or through) f". Similarly, the inverse image (or preimage) of a given subset B of the codomain Y is the set of all elements of X that map to a member of B. The image of the function f is the set of all output values it may produce, that is, the image of X. The preimage of f is the preimage of the codomain Y. Because it always equals X (the domain of f), it is rarely used. Image and inverse image may also be defined for general Binary relation#Operations, binary relations, not just functions. Definition The word "image" is used in three related ways. In these definitions, f : X \to Y is a Function (mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Direct Product
In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abstraction of these notions in the setting of category theory. Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance. The direct sum is a related operation that agrees with the direct product in some but not all cases. Examples * If \R is thought of as the set of real numbers without further structure, the direct product \R \times \R is just the Cartesian product \. * If \R is thought of as the group of real numbers under addition, the direct product \R\times \R still has \ as its underlying set. The difference between this and the preceding examples is that \R \times \R is now a group and so how to add their elements must also be s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Real Closed Ring
In mathematics, a real closed ring (RCR) is a commutative ring ''A'' that is a subring of a product ring, product of real closed fields, which is closed under continuous function, continuous Semialgebraic set, semi-algebraic functions defined over the integers. Examples of real closed rings Since the rigorous definition of a real closed ring is of technical nature it is convenient to see a list of prominent examples first. The following rings are all real closed rings: * real closed fields. These are exactly the real closed rings that are field (mathematics), fields. * the ring of all Tychonoff space#Real-valued continuous functions, real-valued continuous functions on a completely regular space ''X''. Also, the ring of all bounded function, bounded real number, real-valued continuous functions on ''X'' is real closed. * convex subrings of real closed fields. These are precisely those real closed rings which are also valuation rings and were initially studied by Cherlin and Dickm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Localization (commutative Algebra)
In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module ''R'', so that it consists of fractions \frac, such that the denominator ''s'' belongs to a given subset ''S'' of ''R''. If ''S'' is the set of the non-zero elements of an integral domain, then the localization is the field of fractions: this case generalizes the construction of the field \Q of rational numbers from the ring \Z of integers. The technique has become fundamental, particularly in algebraic geometry, as it provides a natural link to sheaf theory. In fact, the term ''localization'' originated in algebraic geometry: if ''R'' is a ring of functions defined on some geometric object (algebraic variety) ''V'', and one wants to study this variety "locally" near a point ''p'', then one considers the set ''S'' of all functions that are not zero at ''p'' and localizes ''R' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |