ideals

Field of sets - Stone representation. Every finite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power set representation can be constructed more generally for any complete atomic Boolean algebra. In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields o ...

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Field of sets, Field of sets - Fields of sets in the representation theory of Boolean algebras, Field of sets - Stone representation, Field of sets - Separative and compact fields of sets: towards Stone duality, Field of sets - Fields of sets with additional structure, Field of sets - Sigma algebras and measure spaces, Field of sets - Topological fields of sets, Field of sets - Preorder fields, Field of sets - Complex algebras and fields of sets on relational structures

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In public discussions, some ideas seem to arise more commonly than others. Indeed, often completely separate people may be found to think alike in startling ways. For social scientists, one way of explaining such instances of common opinion is the presence of an ideology. Every society has an ideology that forms the basis of the "public opinion" or common sense, a basis that usually remains invisible to most people within the society. This dominant ideology appears as "neutral", holding to assumptions that are largely unchallenged. Me ...

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Ideology, Ideology - Ideology in everyday society, Ideology - History of the concept of ideology, Ideology - The analysis of ideology, Ideology - Ideology as an instrument of social reproduction, Ideology - Louis Althusser's Ideological State Apparatuses, Ideology - Feminism as critique of ideology, Ideology - Political ideologies, Ideology - List of political ideologies, Ideology - Epistemological ideologies

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Much of the theory of rings continues to make sense when applied to arbitrary semirings. In particular, one can generalise the theory of algebras over commutative rings directly to a theory of algebras over commutative semirings. Then a ring is simply an algebra over the commutative semiring Z of integers. Some mathematicians go so far as to say that semirings are really the more fundamental concept, and specialising to rings should be seen in the same light as specialising ...

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Semiring, Semiring - Definition, Semiring - Examples, Semiring - Semiring theory, Semiring - Further generalizations

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In general parlance, "idealism" or "idealist" is also used to describe a person having high ideals, sometimes with the connotation that those ideals are unrealisable or at odds with "practical" life. The word "ideal" is commonly used as an adjective to designate qualities of perfection, desirability, and excellence. This is foreign to the epistemological use of the word "idealism" which pertains to internal mental representations. These internal ideas represent objects that ar ...

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Idealism, Idealism - History, Idealism - Idealism in the East, Idealism - Idealism in the West, Idealism - Critique of Idealism, Idealism - Idealism in religious thought, Idealism - Other uses

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Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in additive categories can be characterised by the following biproduct condition: The object B is a biproduct of the objects A1,...,An iff there are projection morphisms pj: B → Aj and injection morphisms ij: Aj → B, such that (See also:

Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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If C and D are preadditive categories, then a functor F: C → D is additive if it too is enriched over the category Ab. That is, F is additive iff, given any objects A and B of C, the function F: Hom(A,B) → Hom(F(A),F(B)) is a group homomorphism. Most fun ...

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Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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One of the most general forms of the Chinese remainder theorem can be formulated for rings and (two-sided) ideals. If R is a ring and I1, ..., Ik are ideals of R which are pairwise coprime (meaning that Ii + Ij = R whenever i ≠ j), then the product I of these ideals is equal to their intersection, and the ring R/I is isomorphic to the product ring R/I1 x R/I2 x ... x R/Ik via t ...

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Chinese remainder theorem, Chinese remainder theorem - Simultaneous congruences of integers, Chinese remainder theorem - Statement for principal ideal domains, Chinese remainder theorem - Statement for general rings, Chinese remainder theorem - Applications of the Chinese remainder theorem

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The prototypical properties that were discussed for Boolean algebras in the above section can easily be modified to include more general lattices, such as distributive lattices or Heyting algebras. However, in these cases maximal ideals are different from prime ideals, and the relation between PITs and MITs is not obvious. Indeed, it turns out that the MITs for distributive lattices and even for Heyting algebras are equivalent to the axiom of choice. On the other hand, it is known that the strong PIT for distributive lattices is equiv ...

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Boolean prime ideal theorem, Boolean prime ideal theorem - Prime ideal theorems, Boolean prime ideal theorem - Boolean prime ideal theorem, Boolean prime ideal theorem - Further prime ideal theorems, Boolean prime ideal theorem - Applications, Boolean prime ideal theorem - Literature

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For a principal ideal domain R the Chinese remainder theorem takes the following form: If u1, ..., uk are elements of R which are pairwise coprime, and u denotes the product u1...uk, then the ring R/uR and the product ring R/u1R x ... x R/ukR are isomorphic via the isomorphism such that The inverse isomorphism can be constructed as follows. For e ...

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Chinese remainder theorem, Chinese remainder theorem - Simultaneous congruences of integers, Chinese remainder theorem - Statement for principal ideal domains, Chinese remainder theorem - Statement for general rings, Chinese remainder theorem - Applications of the Chinese remainder theorem

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The most obvious example of a preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. (Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. In contrast, the category of all groups is not closed.) See medial category. Other common examples: The category of (left) modules over a ring R, in particular: the category of vector spaces over a field K. The algebra of m ...

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Preadditive category, Preadditive category - Examples, Preadditive category - Elementary properties, Preadditive category - Additive functors, Preadditive category - Biproducts, Preadditive category - Kernels and cokernels, Preadditive category - Special cases

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Perhaps the most accessible source for the original meaning of "ideology" is Hippolyte Taine's work on the Ancien Regime (first volume of "Origins of Contemporary France"). He describes ideology as rather like teaching philosophy by the Socratic method, but without extending the vocabulary beyond what the general reader already possessed, and without the examples from observation which practical science would require. Taine identifies it not just with Destutt de Tracy, but wi ...

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Ideology, Ideology - Ideology in everyday society, Ideology - History of the concept of ideology, Ideology - The analysis of ideology, Ideology - Ideology as an instrument of social reproduction, Ideology - Louis Althusser's Ideological State Apparatuses, Ideology - Feminism as critique of ideology, Ideology - Political ideologies, Ideology - List of political ideologies, Ideology - Epistemological ideologies

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Before proceeding to actual prime ideal theorems, recall that an order theoretical ideal is just a (non-empty) directed lower set. If the considered poset has binary suprema like the posets within this article, then this is equivalently characterized as a lower set I which is closed for binary suprema (i.e. x, y in I imply xy in I). An ideal I is prime if, whenever an infimum xy is in I, one also has x in I or y in < ...

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Boolean prime ideal theorem, Boolean prime ideal theorem - Prime ideal theorems, Boolean prime ideal theorem - Boolean prime ideal theorem, Boolean prime ideal theorem - Further prime ideal theorems, Boolean prime ideal theorem - Applications, Boolean prime ideal theorem - Literature

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Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has a root: this is the statement of the fundamental theorem of algebra. There is a difference between approximating roots and finding concrete closed formulas for them. Formulas for the roots of polynom ...

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Polynomial, Polynomial - Elementary properties of polynomials, Polynomial - More advanced examples of polynomials, Polynomial - History, Polynomial - Polynomial functions, Polynomial - Graphs, Polynomial - End behavior, Polynomial - Number of x-intercepts, Polynomial - Number of turning points, Polynomial - Examples, Polynomial - Notes, Polynomial - Roots, Polynomial - Numerical analysis, Polynomial - Polynomials and calculus, Polynomial - Evaluation of polynomials, Polynomial - Finding roots, Polynomial - Several variables, Polynomial - Abstract algebra, Polynomial - Divisibility, Polynomial - More variables

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The idea is generalized in universal algebra: A congruence relation on an algebra A is a subset of the direct product A × A that is both an equivalence relation on A and a subalgebra of A × A. Congruences typically arise as kernels of homomorphisms, and in fact every congruence is the kernel of some homomorphism: For a given congruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra. Furthermore, the function that maps every element of A to its equivalence class is ...

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Congruence relation, Congruence relation - Modular arithmetic, Congruence relation - Linear algebra, Congruence relation - Universal algebra, Congruence relation - Group theory, Congruence relation - Ring theory, Congruence relation - General case of kernels

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Two ideals A and B in the commutative ring R are called coprime if A + B = R. This generalizes Bézout's identity: with this definition, two principal ideals (a) and (b) in the ring of integers Z are coprime if and only if a and b are coprime. If the ideals A and B of R are coprime, then AB = A∩B; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese Remainder Theorem is a ...

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Coprime, Coprime - Properties, Coprime - Cross notation group, Coprime - Generalizations

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Two real matrices A and B are called congruent if there is an invertible real matrix P such that A symmetric matrix has real eigenvalues. The inertia of a symmetric matrix is a triple consisting of the number of positive eigenvalues, the number of zero eigenvalues, and the number of negative eigenvalues. Sylvester's law of inertia states that two symmetric real matrices are congruent if and only if they the same inertia. So, congruence transformations may change the eigenvalues of a matri ...

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Congruence relation, Congruence relation - Modular arithmetic, Congruence relation - Linear algebra, Congruence relation - Universal algebra, Congruence relation - Group theory, Congruence relation - Ring theory, Congruence relation - General case of kernels

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The prototypical example is modular arithmetic: for n a positive natural number, two integers a and b are called congruent modulo n if a − b is divisible by n. If and , then and . This turns the equivalence (mod n) into a congruence on the ring of all integers. ...

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Congruence relation, Congruence relation - Modular arithmetic, Congruence relation - Linear algebra, Congruence relation - Universal algebra, Congruence relation - Group theory, Congruence relation - Ring theory, Congruence relation - General case of kernels

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There are various other mathematical concepts that can be used to represent complete lattices. One means of doing so is the Dedekind-MacNeille completion. When this completion is applied to a poset that already is a complete lattice, then the result is a complete lattice of sets which is isomorphic to the original one. Thus we immediately find that every complete lattice is isomorphic to a complete lattice of sets. Another representation is obtained by noting that the image of any closure operator on a complete lattice is again a comp ...

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Complete lattice, Complete lattice - Formal definition, Complete lattice - Complete semilattices, Complete lattice - Examples, Complete lattice - Morphisms of complete lattices, Complete lattice - Free construction and completion, Complete lattice - Free complete semilattices, Complete lattice - Free complete lattices, Complete lattice - Completion, Complete lattice - Representation, Complete lattice - Further results, Complete lattice - Literature

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Lasker was also a distinguished mathematician. He performed his doctoral studies at Erlangen from 1900 to 1902 under David Hilbert. His doctoral thesis, Über Reihen auf der Convergenzgrenze, was published in Philosophical Transactions in 1901. Lasker introduced the concept of a primary ideal, which extends the notion of a power of a prime number to algebraic geometry. He is most famous for his 1905 paper Zur Theorie der Moduln und Ideale, which appeared in Mathematische Annalen. In this paper, he established what is now known as the Lasker-Noether theorem for ...

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Emanuel Lasker, Emanuel Lasker - Chess champion, Emanuel Lasker - Mathematician, Emanuel Lasker - Other facets of his life, Emanuel Lasker - Books, Emanuel Lasker - Quotations

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He was also a philosopher, and a good friend of Albert Einstein. Later in life he became an ardent humanitarian, and wrote passionately about the need for inspiring and structured education for the stabilization and security of mankind. He also took up bridge and became a master at it, in addition to studying Go. He invented Lasca, a draughts-like game, where instead of removing captured pieces from the board, they are stacked underneath the capturer. The poet Else Lasker-Schüler was his sister-in- ...

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Emanuel Lasker, Emanuel Lasker - Chess champion, Emanuel Lasker - Mathematician, Emanuel Lasker - Other facets of his life, Emanuel Lasker - Books, Emanuel Lasker - Quotations

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The Heian period is preceded by the Nara period and began in 794 after the movement of the capital of Japanese civilization to Heian-kyō (present-day Kyoto) by the 50th emperor, Emperor Kammu. It is considered a high point in Japanese culture that later generations have always admired. Also, the time period is also noted for the rise of the samurai class, which would eventually take power and start the feudal period of Japan. The capital was also named af ...

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Heian period, Heian period - History, Heian period - Developments in Buddhism, Heian period - Heian period literature, Heian period - Heian period economics, Heian period - The Fujiwara Regency, Heian period - The Rise of the military class

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A non-empty subset F of a partially ordered set (P,≤) is a filter, if the following conditions hold: For every x, y in F, there is some element z in F, such that z ≤ x and z ≤ y. (F is a filter base) For every x in F and y in P, x ≤ y implies that y is in F. (F is an upper set) A filter is See also:

Filter mathematics, Filter mathematics - General definition, Filter mathematics - Filter on a set, Filter mathematics - Examples, Filter mathematics - Filters in model theory, Filter mathematics - Filters in topology, Filter mathematics - Filters in uniform spaces

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