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Topic: Complete lattice

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	[No title] (Site not responding. Last check: 2007-10-21)
		This concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.
		More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left-invariant, or right-invariant - the definition is independent of that choice).
		The lattice of submodules of a module and the lattice of normal subgroups of a group have the special property that x v (y ^ (x v z)) = (x v y) ^ (x v z) for all x, y and z in the lattice.
	www.informationgenius.com /encyclopedia/l/la/lattice_1.html (1343 words)

	Numeri reali: Tutte le informazioni su Numeri reali su Encyclopedia.it (Site not responding. Last check: 2007-10-21)
		Prima di tutto, un ordinamento può essere lattice complete.
		È facile vedere che nessun campo ordinato può essere lattice complete, perché può non avere l'elemento massimo (dato un elemento z, z + 1 è più grande).
		Non è vero che R è l'unico campo uniformly complete ordered, ma è l'unico campo ordinato completo dotato della proprietà di Archimede, quindi si può sentire spesso la frase "campo completo di Archimede" invece di "campo ordinato completo".
	www.encyclopedia.it /n/nu/numeri_reali.html (2029 words)

	Order theory Details, Meaning Order theory Article and Explanation Guide
		Lattices, in which every non-empty finite set has a supremum and infimum,
		Directed complete partial orders (dcpos), that guarantee the existence of suprema of all directed subsets and that are studied in domain theory.
		Beyond these relations, topology can be looked at solely in terms of the open set lattices, which leads to the study of pointless topology.
	www.e-paranoids.com /o/or/order_theory.html (4039 words)

	Recursive Functions
		Banach [1922] has proved that a contraction on a complete metric space has a unique fixed point, and the proof is a typical iteration.
		Either the vector vanishes on a point of the border on one of the squares, thus determining a fixed point of the given function, or there is at least one square on which the vector makes a complete turn while the point moves around the border, and the process can be started again.
		Abian and Brown [1961] replaced complete lattices by chain-complete partial orderings, in which every chain of elements has a least upper bound.
	plato.stanford.edu /entries/recursive-functions (6936 words)

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