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Movements, leaders, victories and the continuing fight for equality.
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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
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In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.)
In a partially ordered set (P,≤) an element c is called compact (or finite) if it satisfies one of the following equivalent conditions:
If the poset P additionally is a join-semilattice (i.e., if it has binary suprema) then these conditions are equivalent to the following statement:
In particular, if c = sup S, then c is the supremum of a finite subset of S.
These equivalences are easily verified from the definitions of the concepts involved. For the case of a join-semilattice, any set can be turned into a directed set with the same supremum by closing under finite (non-empty) suprema.
When considering directed complete partial orders or complete lattices the additional requirements that the specified suprema exist can of course be dropped. A join-semilattice that is directed complete is almost a complete lattice (possibly lacking a least element)—see completeness (order theory) for details.
A poset in which every element is the supremum of the directed set formed by the compact elements below it is called an algebraic poset. Such posets that are dcpos are much used in domain theory.
As an important special case, an algebraic lattice is a complete lattice L where every element x of L is the supremum of the compact elements below x.
A typical example (which served as the motivation for the name "algebraic") is the following:
For any algebraic system A (for example, a group, a ring, a field, a lattice, etc.; or even a mere set without any operations), let Sub(A) be the set of all substructures of A, i.e., of all subsets of A which are closed under all operations of A (group addition, ring addition and multiplication, etc.). Here the notion of substructure includes the empty substructure in case the algebra A has no nullary operations.
Then:
Also, a kind of converse holds: Every algebraic lattice is isomorphic to Sub(A) for some algebra A.
There is another algebraic lattice that plays an important role in universal algebra: For every algebra A we let Con(A) be the set of all congruence relations on A. Each congruence on A is a subalgebra of the product algebra AxA, so Con(A) ⊆ Sub(AxA). Again we have
Again there is a converse: By a theorem of George Grätzer and E. T. Schmidt, every algebraic lattice is isomorphic to Con(A) for some algebra A.
Compact elements are important in computer science in the semantic approach called domain theory, where they are considered as a kind of primitive element: the information represented by compact elements cannot be obtained by any approximation that does not already contain this knowledge. Compact elements cannot be approximated by elements strictly below them. On the other hand, it may happen that all non-compact elements can be obtained as directed suprema of compact elements. This is a desirable situation, since the set of compact elements is often smaller than the original poset—the examples above illustrate this.
See the literature given for order theory and domain theory.
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
In the mathematical area of order theory, the compact elements or finite elements of a partially ordered set are those elements that cannot be subsumed by a supremum of any non-empty directed set that does not already contain members above the compact element. This notion of compactness simultaneously generalizes the notions of finite sets in set theory, compact sets in topology, and finitely generated modules in algebra. (There are other notions of compactness in mathematics.)
The Honda Element (Japanese: ホンダ・エレメント, Hepburn: Honda Eremento) is a compact crossover SUV manufactured by Honda and marketed in North America over a single generation for model years 2003–2011, and noted for its boxy exterior styling with bi-parting side doors and its boxy, flexible interior layout. Manufactured in East Liberty, Ohio, the Element used a modified second generation CR-V platform with front-wheel or a system marketed as “real time 4-wheel drive” that sends some torque to the rear wheels if the front wheels lose traction. In late 2010, shortly before its discontinuation, production had totaled approximately 325,000.
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . There is a closely related theorem classifying the irreducible representations of a connected compact Lie group K {\displaystyle K} . The theorem states that there is a bijection λ ↦ [ V λ ] {\displaystyle \lambda \mapsto [V^{\lambda }]} from the set of "dominant integral elements" to the set of equivalence classes of irreducible representations of g {\displaystyle {\mathfrak {g}}} or K {\displaystyle K} . The difference between the two results is in the precise notion of "integral" in the definition of a dominant integral element. If K {\displaystyle K} is simply connected, this distinction disappears. The theorem was originally proved by Élie Cartan in his 1913 paper. The version of the theorem for a compact Lie group is due to Hermann Weyl. The theorem is one of the key pieces of representation theory of semisimple Lie algebras.
In mathematics, especially general topology and mathematical analysis, compactness is a property of a space that makes it behave in many ways like a finite set. For instance, on a finite set every infinite sequence must take some value infinitely often, by the pigeonhole principle. For subsets of Euclidean space, the analogous statement is sequential compactness: a set is compact if and only if every infinite sequence in the set has a subsequence that converges to a point of the set. Likewise, whereas every real-valued function on a finite set is bounded and attains its maximum and minimum, every continuous real-valued function on a compact space has these properties. For compact subsets of Euclidean space, this is the extreme value theorem. Another basic property of finite sets is that every cover of a finite set by subsets has a finite subcover: one may choose, for each point of the finite set, a member of the cover containing it. The corresponding topological property is used to define compactness: a topological space is compact if every open cover has a finite subcover. In metric spaces this is equivalent to several other formulations, including sequential compactness, though these equivalences can fail in more general topological spaces. Thus every sequence in the closed unit interval [0,1] has a convergent subsequence with limit in [0,1], whereas this fails for spaces such as the open interval (0,1) and the real line. For subsets of Euclidean space, compactness is equivalent to being closed and bounded, by the Heine–Borel theorem. The property of compactness often allows local information to be combined into global conclusions. The term compact set may refer either to a compact topological space or, more commonly, to a subset of a topological space that is compact in the subspace topology. Compactness was formally introduced by Maurice Fréchet in 1906 in work generalizing the Bolzano–Weierstrass theorem from sets of points to spaces of functions. Later, Pavel Alexandrov and Pavel Urysohn developed the open-cover formulation that is now standard in topology. Compactness plays a central role throughout mathematics; for example, continuous real-valued functions on compact spaces attain maxima and minima, and major results such as the Arzelà–Ascoli theorem and the Peano existence theorem depend on compactness.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Brown v. Board of Education in 1954.
