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structure

tent structure

Building that uses masts or poles and tensile membrane (e.g., fabric or animal skin) enclosures. Tent structures are prestressed by externally applied forces so that they are held taut under anticipated load conditions. Tents have been the dwelling places of most of the world's nomadic peoples, from ancient times until the present. The traditional Bedouin tent consists of a rectangular membrane of strips of woven camel hair that is strained on webbing straps and secured with guys over a rectangle of poles. The American Plains Indians developed the conical tepee. The Central Asian nomadic pole dwelling, or yurt, uses skins and textiles as its covering. Seealso membrane structure, pole construction.

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Several methods of representing a molecule's structure. In Lewis structures, element symbols elipsis

Smallest identifiable unit into which a pure substance can be divided and retain its composition and chemical properties. Division into still smaller parts, eventually atoms, involves destroying the bonding that holds the molecule together. For noble gases, the molecule is a single atom; all other substances have two (diatomic) or more (polyatomic) atoms in a molecule. The atoms are the same in elements, such as hydrogen (H2), and different in compounds, such as glucose (C6H12O6). Atoms always combine into molecules in fixed proportions. Molecules of different substances can have the same constituent atoms, either in different proportions, as in carbon monoxide (CO) and carbon dioxide (CO2), or bonded in different ways (see isomer). The covalent bonds in molecules give them their shapes and most of their properties. (The concept of molecules has no significance in solids with ionic bonds.) Analysis with modern techniques and computers can determine and display the size, shape, and configuration of molecules, the positions of their nuclei and electron clouds, the lengths and angles of their bonds, and other details. Electron microscopy can even produce images of individual molecules and atoms. Seealso molecular weight.

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or frame structure

Structure supported mainly by a skeleton, or frame, of wood, steel, or reinforced concrete rather than by load-bearing walls. Rigid frames have fixed joints that enable the frames to resist lateral forces; other frames require diagonal bracing or shear walls and diaphragms for lateral stability. Heavy timber framing was the most common type of construction in East Asia and northern Europe from prehistoric times to the mid-19th century. It was supplanted by the balloon frame and the platform frame (see light-frame construction). Steel's strength, when used in steel framing, made possible buildings with longer spans. Concrete frames impart greater rigidity and continuity; various advancements, such as the introduction of the shear wall and slipforming, have made concrete a serious competitor with steel in high-rise structures.

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Rock crystal from the Dauphiné region of France.

Transparent variety of the silica mineral quartz that is valued for its clarity and total lack of colour or flaws. Rock crystal formerly was used extensively as a gemstone, but it has been replaced by glass and plastic; rhinestones originally were quartz pebbles found in the Rhine River. The optical properties of rock crystal led to its use in lenses and prisms; its piezoelectric properties (see piezoelectricity) are used to control the oscillation of electrical circuits.

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Optoelectronic device used in displays for watches, calculators, notebook computers, and other electronic devices. Current passed through specific portions of the liquid crystal solution causes the crystals to align, blocking the passage of light. Doing so in a controlled and organized manner produces visual images on the display screen. The advantage of LCDs is that they are much lighter and consume less power than other display technologies (e.g., cathode-ray tubes). These characteristics make them an ideal choice for flat-panel displays, as in portable laptop and notebook computers.

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Substance that flows like a liquid but maintains some of the ordered structure characteristic of a crystal. Some organic substances do not melt directly when heated but instead turn from a crystalline solid to a liquid crystalline state. When heated further, a true liquid is formed. Liquid crystals have unique properties. The structures are easily affected by changes in mechanical stress, electromagnetic fields, temperature, and chemical environment. Seealso liquid crystal display.

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Any solid material whose atoms are arranged in a definite pattern and whose surface regularity reflects its internal symmetry. Each of a crystal's millions of individual structural units (unit cells) contains all the substance's atoms, molecules, or ions in the same proportions as in its chemical formula (see formula weight). The cells are repeated in all directions to form a geometric pattern, manifested by the number and orientation of external planes (crystal faces). Crystals are classified into seven crystallographic systems based on their symmetry: isometric, trigonal, hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic. Crystals are generally formed when a liquid solidifies, a vapour becomes supersaturated (see saturation), or a liquid solution can no longer retain dissolved material, which is then precipitated. Metals, alloys, minerals, and semiconductors are all crystalline, at least microscopically. (A noncrystalline solid is called amorphous.) Under special conditions, a single crystal can grow to a substantial size; examples include gemstones and some artificial crystals. Few crystals are perfect; defects affect the material's electrical behaviour and may weaken or strengthen it. Seealso liquid crystal.

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The Crystal Palace at Sydenham Hill, London. It was designed by Sir Joseph Paxton for the Great elipsis

Giant glass-and-iron exhibition hall in Hyde Park, London, that housed the Great Exhibition of 1851. It was taken down and rebuilt (1852–54) at Sydenham Hill, where it survived until its destruction by fire in 1936. Designed by the greenhouse builder Sir Joseph Paxton (1801–1865), it was a remarkable assembly of prefabricated parts. Its intricate network of slender iron rods sustaining walls of clear glass established an architectural standard for later international exhibitions, likewise housed in glass conservatories.

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The classical “planetary” model of an atom. The protons and neutrons in the nucleus are elipsis

Smallest unit into which matter can be divided and still retain the characteristic properties of an element. The word derives from the Greek atomos (“indivisible”), and the atom was believed to be indivisible until the early 20th century, when electrons and the nucleus were discovered. It is now known that an atom has a positively charged nucleus that makes up more than 99.9percnt of the atom's mass but only about 1/100,000 of its volume. The nucleus is composed of positively charged protons and electrically neutral neutrons, each about 2,000 times as massive as an electron. Most of the atom's volume consists of a cloud of electrons that have very small mass and negative charge. The electron cloud is bound to the nucleus by the attraction of opposite charges. In a neutral atom, the protons in the nucleus are balanced by the electrons. An atom that has gained or lost electrons becomes negatively or positively charged and is called an ion.

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In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets.

Two equivalent definitions

A Boolean algebra is a six-tuple consisting of a set A, equipped with two binary operations land (called "meet" or "and"), lor (called "join" or "or"), a unary operation lnot (called "complement" or "not") and two elements 0 and 1 (sometimes denoted by ⊥ and top), such that for all elements a, b and c of A, the following axioms hold:

a ∨ (bc) = (ab) ∨ c a ∧ (bc) = (ab) ∧ c associativity
ab = ba ab = ba commutativity
a ∨ (ab) = a a ∧ (ab) = a absorption
a ∨ (bc) = (ab) ∧ (ac)   a ∧ (bc) = (ab) ∨ (ac)   distributivity
a ∨ ¬a = 1 a ∧ ¬a = 0 complements

A Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. (Some authors require 0 and 1 to be distinct elements in order to exclude this case.)

It follows from the first three pairs of axioms above (associativity, commutativity and absorption) that

a = ab     if and only if     ab = b.
The relation ≤ defined by ab if and only if the above equivalent conditions hold, is a partial order with least element 0 and greatest element 1. The meet ab or the join ab of two elements coincides with their infimum or supremum, respectively, with respect to ≤.

As in every lattice, the relations ∧ and ∨ satisfy the first three pairs of axioms above; the fourth pair is just distributivity. Since the complements in a distributive lattice are unique, to define the involution ¬ it suffices to define ¬a as the complement of a.

The set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore by applying this operation to a Boolean algebra (or Boolean lattice), one obtains another Boolean algebra with the same elements; it is called its dual.

Examples

  • The simplest non-trivial Boolean algebra has only two elements, 0 and 1, and is defined by the rules:

0 1
0 0 0
1 0 1
0 1
0 0 1
1 1 1
a 0 1
¬a 1 0

* It has applications in logic, interpreting 0 as false, 1 as true, ∧ as and, ∨ as or, and ¬ as not. Expressions involving variables and the Boolean operations represent statement forms, and two such expressions can be shown to be equal using the above axioms if and only if the corresponding statement forms are logically equivalent.

* The two-element Boolean algebra is also used for circuit design in electrical engineering; here 0 and 1 represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, and two such expressions are equal for all values of the variables if and only if the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression.

* The two-element Boolean algebra is also important in the general theory of Boolean algebras, because an equation involving several variables is generally true in all Boolean algebras if and only if it is true in the two-element Boolean algebra (which can always be checked by a trivial brute force algorithm). This can for example be used to show that the following laws (Consensus theorems) are generally valid in all Boolean algebras:
** (ab) ∧ (¬ac) ∧ (bc) ≡ (ab) ∧ (¬ac)
** (ab) ∨ (¬ac) ∨ (bc) ≡ (ab) ∨ (¬ac)

  • Starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra (that is, the set of sentences in the propositional calculus modulo tautology). This construction yields a Boolean algebra. It is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to {0,1}.
  • Given any linearly ordered set with a least element, the interval algebra is the smallest algebra of subsets of L containing all of the half-open intervals [a, b) such that a is in L and b in either in L or equal to ∞. Interval algebras are useful in the study of Lindenbaum-Tarski algebras; every countable BA is isomorphic to an interval algebra.
  • The power set (set of all subsets) of any given nonempty set S forms a Boolean algebra with the two operations ∨ := ∪ (union) and ∧ := ∩ (intersection). The smallest element 0 is the empty set and the largest element 1 is the set S itself.
  • The set of all subsets of S that are either finite or cofinite is a Boolean algebra.
  • For any natural number n, the set of all positive divisors of n forms a distributive lattice if we write ab for a | b. This lattice is a Boolean algebra if and only if n is square-free. The smallest element 0 of this Boolean algebra is the natural number 1; the largest element 1 of this Boolean algebra is the natural number n.
  • Other examples of Boolean algebras arise from topological spaces: if X is a topological space, then the collection of all subsets of X which are both open and closed forms a Boolean algebra with the operations ∨ := ∪ (union) and ∧ := ∩ (intersection).
  • If R is an arbitrary ring and we define the set of central idempotents by
    A = { eR : e2 = e, ex = xe, ∀xR }
    then the set A becomes a Boolean algebra with the operations ef := e + fef and ef := ef.

Homomorphisms and isomorphisms

A homomorphism between two Boolean algebras A and B is a function f : AB such that for all a, b in A:

f(a lor b) = f(a) lor f(b)
f(a land b) = f(a) land f(b)
f(0) = 0
f(1) = 1

It then follows that fa) = ¬f(a) for all a in A as well. The class of all Boolean algebras, together with this notion of morphism, forms a full subcategory of the category of lattices.

A homomorphism is called a monomorphism, epimorphism or isomorphism if it is injective, surjective or bijective, respectively. The inverse map of an isomorphism is also an isomorphism.

Boolean rings, ideals and filters

Every Boolean algebra (A, land, lor) gives rise to a ring (A, +, *) by defining a + b = (a land ¬b) lor (b land ¬a) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a * b = a land b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a * a = a for all a in A; rings with this property are called Boolean rings.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x lor y = x + y - xy and x land y = xy. Since these two operations are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : AB is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x lor y in I and for all a in A we have a land x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if IA and if a land b in I always implies a in I or b in I. An ideal I of A is called maximal if IA and if the only ideal properly containing I is A itself. These notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A.

The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x land y in p and for all a in A if a lor x = a then a in p.

Representing Boolean algebras

It can be shown that every finite Boolean algebra is isomorphic to the Boolean algebra of all subsets of a finite set. Therefore, the number of elements of every finite Boolean algebra is a power of two.

Stone's celebrated representation theorem for Boolean algebras states that every Boolean algebra A is isomorphic to the Boolean algebra of all closed-open sets in some (compact totally disconnected Hausdorff) topological space.

Axiomatics for Boolean algebras

Let the unary functional symbol n be read as 'complement'. In 1933, the American mathematician Edward Vermilye Huntington (1874–1952) set out the following elegant axiomatization for Boolean algebra:

  1. Commutativity: x + y = y + x.
  2. Associativity: (x + y) + z = x + (y + z).
  3. Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x.

Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:

4. Robbins Equation: n(n(x + y) + n(x + n(y))) = x,

do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question remained open for decades, and became a favorite question of Alfred Tarski and his students.

In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the automated reasoning program EQP he designed. For a simplification of McCune's proof, see Dahn (1998).

Generalizations

Removing the requirement of existence of a unit from the axioms of Boolean algebra yields "generalized Boolean algebras". Formally, a distributive lattice B is a generalized Boolean lattice, if it has a smallest element 0 and for any elements a and b in B such that ab, there exists an element x such that aland x=0 and alor x=b. Defining asetminus b as the unique x such that (aland b)lor x=a and (aland b)land x=0, we say that the structure (B,land,lor,setminus,0) is a generalized Boolean algebra, while (B,lor,0) is a generalized Boolean semilattice. Generalized Boolean lattices are exactly the ideals of Boolean lattices.

A structure that satisfies all axioms for Boolean algebras except the two distributivity axioms is called an orthocomplemented lattice. Orthocomplemented lattices arise naturally in quantum logic as lattices of closed subspaces for separable Hilbert spaces.

History

The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English mathematician. He introduced the algebraic system of logic initially in a small pamphlet, The Mathematical Analysis of Logic, published in 1847 in response to an ongoing public controversy between Augustus De Morgan and William Hamilton, and later as a more substantial book, The Laws of Thought, published in 1854. Boole's formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a dual pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons and Charles Peirce. To the 1890 Vorlesungen of Ernst Schröder we owe the first systematic presentation of Boolean algebra and distributive lattices. The first extensive treatment of Boolean algebra in English is A. N. Whitehead's 1898 Universal Algebra. Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward Vermilye Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoff's 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing and Boolean-valued models.

See also

References

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