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Topic: Lattice (order)

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Semilattice

Heyting algebra

Supremum

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Hasse diagram

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	Lattice (order) - Wikipedia, the free encyclopedia
		The set of compact elements of an arithmetic complete lattice is a lattice with a least element, where the lattice operations are given by restricting the respective operations of the arithmetic lattice.
		For example, the lattice of submodules of a module, and the lattice of normal subgroups of a group, all have this special property.
		These conditions basically amount to saying that there is a functor from the category of sets and functions to the category of lattices and lattice homomorphisms which is left adjoint to the forgetful functor from lattices to their underlying sets.
	en.wikipedia.org /wiki/Lattice_(order) (2421 words)

	Lattice
		Every lattice can be generated from a basis for the underlying vector space by considering all linear combinations with integral coefficients.
		In another mathematical usage, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound and a greatest lower bound (also called supremum and infimum, respectively).
		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.ebroadcast.com.au /lookup/encyclopedia/la/Lattice.html (913 words)

	Encyclopedia :: encyclopedia : Lattice (order) (Site not responding. Last check: 2007-09-19)
		In mathematics, a lattice is a partially ordered set (or poset), in which all nonempty finite subsets have both a supremum (join) and an infimum (meet).
		Lattices constitute one of the most prominent representatives of a series of "lattice-like" structures which admit order-theoretic as well as algebraic descriptions, such as semilattices, Heyting algebras, and Boolean algebras.
		The natural numbers in their common order are a lattice, the lattice operations given by the min and max operations.
	www.hallencyclopedia.com /Lattice_(order) (2277 words)

	Lattice (order)
		In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum (join) and an infimum (meet).
		Conversely, the order induced by the algebraically defined lattice (L,,) that was derived from the order theoretic formulation above coincides with the original ordering of L.
		The natural numbers in their common order are a lattice.
	www.brainyencyclopedia.com /encyclopedia/l/la/lattice__order_.html (2552 words)

	Lattice Corporate Profile (Site not responding. Last check: 2007-09-19)
		Lattice's mission is to improve healthcare through the use of handheld technology that allows providers to streamline workflow, reduce paperwork, eliminate preventable medical errors, and improve data collection.
		Lattice is widely recognized as the first company to provide the C programming language on the IBM PC in 1982.
		Lattice was the first company to introduce a library of C routines to access and update dBASE database files, enabling C programmers to utilize data in dBASE files at a time when dBASE was a very popular programming language.
	www.lattice.com /corp.htm (600 words)

	[No title]
		Curiously, a lattice can also be defined in algebra-style, as a set with two operations, Meet and Join, which have certain properties such as commutation, and the algebra definition is equivalent to the ordering one.
		To implement the idea, define a lattice as: A LATTICE is a set such that for every pair of elements x and y in the set, Join(x,y) and Meet (x,y) are also elements of the set.
		In other words, a lattice is a set that includes its least upper and greatest lower bounds, and that in a special sense does not have any holes in it-- because it must include not only its grand meet and join, but every other meet and join of its elements too.
	www.rasmusen.org /GI/lattice.theory.notes.txt (3223 words)

	Wikinfo \| Lattice
		In the first case, geometric pictures of the simplest lattices look like lattices in the sense defined by the dictionary; in the second case, the Hasse diagrams of the posets look (in some simple cases) like the aforementioned lattices.
		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).
	www.wikinfo.org /wiki.php?title=Lattice (1547 words)

	[No title] (Site not responding. Last check: 2007-09-19)
		Between the ordered and the disordered phase(s) free energy contours occur, whose probabilities are estimated to be uniformly small via a contour estimate valid over a whole temperature interval.
		For lattice gauge models the variables are elements of a unitary representation of a compact continuous gauge group, e.g.
		The correspondence, as in \cite{EntShl}, is that ${\mathbb{\varepsilon}}$ is of order $O(\frac {1}{\sqrt p})$.
	www.ma.utexas.edu /mp_arc/papers/03-278 (2827 words)

	Multiplication of Sets
		A space on which two operations are defined in a way that reminds us of the intersection and union of sets is known as a lattice.
		Lattices have been introduced by the German mathematician J.W.R.Dedekind(1831-1916) along with his invention of ideals in rings.
		The word "lattice" was first circulated by the american G.D.Birkhoff (1884-1944) in 1930s.
	www.cut-the-knot.org /do_you_know/mul_set.shtml (415 words)

	Help Texts for posets 2.3
		In order to force the use of the lexordering induced by a particular linear extension of P, the vertex set X may be specified as a list in the desired order, rather than as a set.
		J(P,X) returns (the covering relation of) an abstract poset that is isomorphic to the lattice of order ideals of P. The vertex set of the output is {1,2,...,m} for some m, and corresponds to the listing order used by 'ideals'.
		In order to force a specific choice for the linear extension, the vertex set X may be specified as a list in the desired order, rather than as a set.
	www.math.lsa.umich.edu /~jrs/software/posetshelp.html (6627 words)

	[No title]
		In contrast, for the standard versions of the liquid crystal and lattice gauge models, as well as for very non-linear ferromagnetic $\sigma$-models, both numerics and high temperature series suggested the existence of 1st order transitions, despite some theoretical and numerical analyses originally either suggesting 2nd order transitions, no transitions at all, or Kosterlitz-Thouless type transitions.
		The wide occurrence of first-order transitions in liquid-crystal and lattice gauge models indicates that a proof in these type of models may be of even more direct physical relevance than in the case of ferromagnets.
		For lattice gauge models on the other hand, also the standard (p=1) actions lead to first-order transitions in mean-field theory (\cite{Zin}, section 34.4), which indicates a first-order transition in sufficiently high dimension.
	www.ma.utexas.edu /mp_arc/papers/04-121 (4645 words)

	The Math Forum - Math Library - Order/Lattices (Site not responding. Last check: 2007-09-19)
		A lattice is an infinite arrangement of points spaced with sufficient regularity that one can shift any point onto any other point by some symmetry of the arrangement.
		More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact).
		Ordered sets, or lattices, give a uniform structure to, for example, the set of subfields of a field.
	mathforum.org /library/topics/lattices (1129 words)

	FPGA-Based FIR Filters using Distributed Arithmetic
		Cascade and lattice structures present several interesting properties such as low quantification error and high-stability in the filter coefficients.
		The goal of this article is to implement FPGA-based direct-form, cascade, and lattice high-order FIR filters using bit-serial DA.
		Furthermore, as a result of the four-input LUT structures, the partition of the memories in the FPGA case is limited by r<T/4 (T is the order of the filter).
	www.techonline.com /community/related_content/20135?print (1291 words)

	Amazon.com: Introduction to Lattices and Order: Books: B. A. Davey,H. A. Priestley (Site not responding. Last check: 2007-09-19)
		Lattices are also considered as algebraic structures and their study from this viewpoint reinforces ideas encountered in the theory of groups and rings.
		The representation of distributive lattices by ordered topological spaces is presented; a self-contained treatment of the requisite topology is included.
		Order, order, order - it permeates mathematics, and everyday life, to such an extent that we take it for granted.
	www.amazon.com /exec/obidos/tg/detail/-/0521367662?v=glance (1295 words)

	PRB 64, 024518 (2001)
		For weak pinning the vortex lattice is defect free until the vortex-vortex interactions have been reduced to a low value, when defects begin to appear with a simultaneous rapid increase in the critical depinning force.
		A key reason for this uncertainty is the absence of a systematic connection between the number of defects in the vortex lattice, the softness of the lattice, and the critical current.
		The vortex lattice is relatively ordered in (a).
	www.t12.lanl.gov /home/olson/Papers/shobo.html (2531 words)

	Direct X-Ray Observation of a Single Hexagonal Myofilament Lattice in Native Myofibrils of Striated Muscle -- Iwamoto ...
		The density distribution in the hexagonal myofilament lattice was reconstructed by performing Fourier synthesis by using the
		The reflection seen below the central beamstop (close to the 2.0 reflection in E and close to 1.0 reflection in F) is the total reflection from the glass strip, as the myofibril lay close to the glass surface.
		lattice, indicating that 40% of the x-ray path in the specimen
	www.biophysj.org /cgi/content/full/83/2/1074 (4597 words)

	Computation of Higher-Order Symmetries for Nonlinear Evolution and Lattice Equations -- from Mathematica Information ...
		A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented.
		The scaling properties of the evolution equations and lattice equations are used to determine the polynomial form of the higher-order symmetries.
		Likewise, lattices must be of first order in time but may involve arbitrary shifts in the discretized space variable.
	library.wolfram.com /infocenter/Articles/1773 (202 words)

	Lattice - Wikipedia, the free encyclopedia
		Lattice (order), a type of partially ordered set
		Bravais lattice, 14 possible arrangements of repeating points in 3-D
		Lattice model (physics), a model defined not on a continuum, but on a lattice
	en.wikipedia.org /wiki/Lattice (125 words)

	DisorderDrivenMelting
		In pure single crystal samples of the high Tc superconductors, the vortex line lattice present at low magnetic fields melts into a vortex line liquid via a sharp first order phase transition, as the temperature is increased.
		As the applied magnetic field H, and hence the vortex line density, increases, the random pinning energy per unit volume increases and the disorder becomes a relevent factor in determining the nature of the vortex phase diagram.
		Above this "upper critical point" resistivity vanishes continuously as temperature is decreased; it is unclear if there exists a distinct second order transition to a "vortex glass" phase, or just a cross-over where the vortex liquid viscosity increases sufficiently rapidly so as to make any vortex motion undetectable.
	spider.pas.rochester.edu /M/teitel/DisorderDrivenMelting.html (354 words)

	OpenGL Lattice 3D Screen Saver
		Lattice 3D is an OpenGL screen saver that draws 3D latticework on your desktop when your computer is idle.
		Lattice 3D Screen Saver is an OpenGL application and requires OpenGL library to be installed on your computer (opengl.dll file in your Windows system directory).
		Lattice 3D Screen Saver is try-before-you-buy-software (shareware) and you can evaluate it for free.
	www.gracebyte.com /lattice (306 words)

	4 Lattice Filters
		Lattice structures are widely used in prediction applications.
		The set of PARCOR coefficients for an M-stage lattice predictor are related to the coefficients of the transversal predictor of the same order (see Section 2.4.3).
		Although the operation of lattice filters are usually described in the prediction context, the application of lattice filters is not limited to prediction applications.
	www.dspalgorithms.com /aspt/asptnode23.html (443 words)

	DIMACS Working Group on Order-theoretic Aspects of Epidemiology
		This working group will study how fundamental order-theoretic concepts of TCS and DM such as semiorders, interval orders, general partial orders, and lattices [Fishburn (1985), Trotter (1992)] can be used to improve the results of epidemiological investigations.
		The point lattice construction has found uses in epidemiology through visualizing the relationships of all possible contingency tables to various statistics, effect measures, and cut-off choices (see, e.g., [Ozonoff and Webster (1997)]) and has also been used in statistics (see [Narayana (1979)]).
		We will build upon a large literature in TCS dealing with order relations, computing them, approximating them, visualizing them, and assigning measures to them, but we will not be building upon a large body of work connecting these ideas to epidemiology, mostly upon the view of several active epidemiologists that these ideas are relevant.
	dimacs.rutgers.edu /Workshops/WGOrder/announcement.html (927 words)

	Introduction
		In particular, the partially ordered set of all submodules of a module forms a (complete) modularlattice; even more general, any sublattice of the lattice of all normal subgroups of a group is a modular lattice.
		This is the generalization of the fundamental theorem of projective geometry* and hence, the representation of mappings between submodule lattices.
		Descriptions of certain isomorphisms betweenisomorphismsof submodule lattices submodule lattices of various classes of modules were given by Von Neumann [1960], Skornjakov [1960], Faltings [1975] and Brehm [1983] and led to semilinear isomorphisms between the underlying modules.
	www1.elsevier.com /homepage/saj/504595/21a.htm (1128 words)

	Visualization of the first-order vortex-lattice melting transition
		In order to improve the sensitivity of the MO technique we have developed a differential method.
		The applied magnetic field and temperature are initially fixed at some value and a first set of images is acquired and averaged.
		This process is repeated typically 100 times and averaged in order to increase the signal to noise ratio.
	www.weizmann.ac.il /home/fnsup/research-mo-visualiz.html (1250 words)

	Lattice
		These subsets form a special kind of partial order that is referred to as a lattice.
		Notice further that the set A is a subset of D and of E; B a subset of D and F and so on.
		The subset relation is, of course, a transitive relation and this property is often exploited to simplify the representation of a lattice.
	www.rci.rutgers.edu /~cfs/472_html/Learn/SimpleLattice_472.html (464 words)

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