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	Lattice (order) - Wikipedia, the free encyclopedia
		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 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.
		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) (2282 words)

	Lattice (order) - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-10-08)
		In mathematics, a lattice is a partially ordered set in which all nonempty finite subsets have both a supremum (join) and an infimum (meet).
		On the other hand, lattices can also be characterized as algebraic structures that satisfy certain identities.
		Consider an algebraic structure in the sense of universal algebra, given by (L,,), where and are two binary operations.
	encyclopedia.learnthis.info /l/la/lattice__order_.html (2515 words)

	Lattice (order) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		The (The number 1 and any other number obtained by adding 1 to it repeatedly) natural numbers in their common order are a lattice, the lattice operations given by the min and max operations.
		Lattices and their homomorphisms obviously form a (A general concept that marks divisions or coordinations in a conceptual scheme) category.
		Other important notions in lattice theory are (The idea of something that is perfect; something that one hopes to attain) ideal and its dual notion (Device that removes something from whatever passes through it) filter.
	www.absoluteastronomy.com /encyclopedia/l/la/lattice_(order).htm (2598 words)

	Scott domain - Wikipedia, the free encyclopedia
		Scott domains are very closely related to algebraic lattices, being different only in possibly lacking a greatest element.
		It is also algebraic, since every finite word happens to be compact and we certainly can approximate infinite words by chains of finite ones.
		Thus this is a Scott domain which is not an algebraic lattice.
	en.wikipedia.org /wiki/Scott_domain (561 words)

	Lattice (order) - the free encyclopedia (Site not responding. Last check: 2007-10-08)
		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, or
		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.
		algebraic lattices, for which the compacts do only form a join-semilattice.
	www.aaez.biz /?t=Lattice_(order) (2113 words)

	Current research (Site not responding. Last check: 2007-10-08)
		The study of residuated lattices is pivotal, because it connects different mathematical viewpoints and paradigms, as well as, diverse mathematical disciplines.
		It is interesting that residuated lattices, structures with prominent algebraic content and history, turn out to constitute algebraic semantics for substructural logics, a fact that establishes a bi-directional connection between logic and algebra.
		It is shown that the generating algebras are in a bijective correspondence to the mechanical bi-infinite words with irrational slope.
	math.vanderbilt.edu /~ngalatos/CurrentResearch.htm (900 words)

	Lattice_(order)
		Consider an algebraic structure in the sense of universal algebra, given by (L, $\vee, \wedge), where \vee and \wedge are two binary operations.$
		This lattice has the empty set as least element, but it will only contain a greatest element if A itself is finite.
		algebraic structures with the two binary operations $\vee and \wedge and the two constants (nullary operations) 0 and 1.$
	www.freecaviar.com /search.php?title=Lattice_(order) (2466 words)

	Creation of Lattices
		Constructs the coordinate lattice with Gram matrix equal to that of L scaled by the integer or rational n.
		Construct the lattice L subseteq Z^n which consists of all vectors reducing modulo p to a code word in C and whose coordinate sum is 0 modulo p^2.
		For n = 16 this is the Barnes-Wall lattice, for n = 24 the Leech lattice.
	www.math.lsu.edu /magma/text804.htm (2105 words)

	Introduction (Site not responding. Last check: 2007-10-08)
		Unfortunately, however, his approach did not allow the formulation of algebraic counterparts of all the geometrical properties he postulated, and therefore the class of modules he used in his characterization is difficult to describe.
		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.
	www.elsevier.com /homepage/saj/504595/21a.htm (1128 words)

	06: Order, lattices, ordered algebraic structures
		Various special types of lattices have particularly nice structure and have applications in group theory and algebraic topology, for example.
		Boolean algebra is used in circuit design and pattern matching; see 94: Information and Circuits and 68: Computer Science.
		Lattices in the sense of section 06 are essentially unrelated to the lattices of number theory.
	www.math.niu.edu /~rusin/known-math/index/06-XX.html (606 words)

	Universal Algebra and Logic Seminar (Site not responding. Last check: 2007-10-08)
		The proof utilizes a construction used in the proof of the corresponding result for residuated lattices and is based on the fact that every residuated lattice with greatest element can be associated in a canonical way with an involutive residuated lattice.
		We will show that for algebraic lattices, PCC is equivalent to the property that the meet of the pseudo-prime elements is zero.
		Abstract: We present sufficient conditions for a residuated lattice (RL) to be represented as an RL of residuated maps on a chain.
	math.vanderbilt.edu /~mmaroti/2004AL (1204 words)

	Institut AIFB - Publikation: A categorical view on algebraic lattices in formal concept analysis (Site not responding. Last check: 2007-10-08)
		In this paper, we explore the notion of algebraicity in formal concept analysis from a category-theoretical perspective.
		To this end, we build on the the notion of approximable concept with a suitable category and show that the latter is equivalent to the category of algebraic lattices.
		At the same time, the paper provides a relatively comprehensive account of the representation theory of algebraic lattices in the framework of Stone duality, relating well-known structures such as Scott information systems with further formalisms from logic, topology, domains and lattice theory.
	www.aifb.uni-karlsruhe.de /Publikationen/showPublikation?publ_id=786 (183 words)

	Lattice Codes for Channel Coding
		Algebraic lattices, i.e., lattices constructed by the canonical embedding of an algebraic number field, provide an efficient tool for designing such codes, since the design criteria are related to properties of the underlying number field.
		Initial search for fully diverse lattices with no restriction on the shape of the lattice resulted in either complex bit labelling procedure or loss in the average energy.
		Algebraic number theory is gaining an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.
	alg-geo.epfl.ch /~foggier/Zn_lattice.html (434 words)

	domain theory (Site not responding. Last check: 2007-10-08)
		From the viewpoint of denotational semantics, algebraic posets are particularly well-behaved, since they allow for the approximation of all elements even when restricting to finite ones.
		Adding even further completeness properties one obtains continuous lattices and algebraic lattices, which are just complete lattices with the respective properties.
		For the algebraic case, one finds broader classes of posets which are still worth studying: historically, the Scott domains were the first structures to be studied in domain theory.
	www.33beat.com /domain_theory.html (2140 words)

	\bf The Duality Between Aglebraic Posets and Bialgebraic Frames: A Lattice Theoretic Perspective
		Bialgebraic (that is, algebraic and dually algebraic), distributive lattices should be familiar enough to algebraists, but this may not be the case with domains.
		Coherent lattices form an important class of mathematical objects, due to their role in Stone-type dualities and to the fact that they possess logical presentations involving only finite conjunctions and disjunctions (see Johnstone [13]).
		This terminology may be traced to the well-known fact that the open-set lattice for the prime spectrum of a bounded, distributive lattice is a coherent frame (see Johnstone [13]).
	frank.mtsu.edu /~jhart/ALGFRM.html (9751 words)

	The Math Forum - Math Library - Order/Lattices (Site not responding. Last check: 2007-10-08)
		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).
		A catalogue of different types of lattices, combinatorial objects with direct applications to packing spheres into a limited amount of space.
	mathforum.org /library/topics/lattices (1109 words)

	Amazon.ca: Books: Semigroups and Their Subsemigroup Lattices (Site not responding. Last check: 2007-10-08)
		The study of various interrelations between algebraic systems and their subsystem lattices is an area of modern algebra which has enjoyed much progress in the recent past.
		Investigations are concerned with different types of algebraic systems such as groups, rings, modules, etc. In semigroup theory, research devoted to subsemigroup lattices has developed over more than four decades, so that much diverse material has accumulated.
		Part A treats semigroups with certain types of subsemigroup lattices, while Part B is concerned with properties of subsemigroup lattices.
	www.amazon.ca /exec/obidos/ASIN/0792342216 (342 words)

	Direct Decompositions of Atomistic Algebraic Lattices - Libkin (ResearchIndex)
		Abstract: this paper we prove the existence of such decompositions for two classes of lattices.
		0.2: The Lattice of Subsemilattices of a Semilattice - Leonid Libkin Department (1994)
		2 the structure of lattices in which every element is a join o..
	citeseer.ist.psu.edu /libkin95direct.html (268 words)

	Lattices: Construction and Operations
		Dual of a lattice, dual quotient of a lattice
		Several interesting lattices are directly accessible inside Magma using standard constructions, e.g., root lattices and preimages of linear codes.
		For each lattice, a LLL-reduced basis for the lattice is computed and stored internally when necessary and subsequently used for many operations.
	magma.maths.usyd.edu.au /magma/Features/node167.html (173 words)

	Universal Algebra and Logic Seminars
		Abstract: A quasivariety is a class of algebras closed under taking products embedded subalgebras and ultraproducts, or equivalently, a class of algebras defined by a set of quasiequations.
		Title: Equivalence of logical consequence operations: an order theoretic perspective, V. Abstract: The aim of this series of talks is to propose and study an order theoretic and categorical framework for the study of the general notion of an abstract logical equivalence between two logical consequence relations.
		These can be used to describe (possibly infinitary) algebras where the only operations are taking meets and joins of selected sets.
	sitemason.vanderbilt.edu /page/f0Cdhu (2029 words)

	Scott is Phoa, locally
		In a paper circulated since end of last year Dana Scott has proposed a category PEQU of algebraic lattices with partial equivalence relations as objects and equivariant Scott continuous maps as morphisms.
		One nice thing about PEQU is that Scott has shown it to be equivalent to the category EQU of T_0 spaces with (total) equivalence relations and equivariant continuous maps.
		For any cardinal $\kappa$ we may consider $\kappa$-PEQU (where the bases of the algebraic lattices are required to have cardinality less or equal $\kappa$) and show it to be equivalent to PER(P $\kappa$) by the same argument as above.
	www.cis.upenn.edu /~bcpierce/types/archives/1997-98/msg00098.html (402 words)

	J. Carmelo Interlando's Publications (Site not responding. Last check: 2007-10-08)
		On the relationship between the diophantine equations over the ring of algebraic integers and the mapping by set partitioning, in Coded Modulation and Bandwidth Efficient Transmission, E. Biglieri and M. Luise (Editors), 1992, Elsevier Science Publishing Co., pp.
		Lattice constellations and codes from quadratic number fields, IEEE Transactions on Information Theory, vol.
		On the optimality of finite constellations from algebraic lattices, Proceedings of the International Workshop on Coding and Cryptography, Paris, France, January 8-12, 2001, pp.
	www.nd.edu /~jinterla/publications.html (768 words)

	Four and Six-Dimensional Signal Constellations From Algebraic Lattices (ResearchIndex)
		Abstract: In this work we describe a procedure to construct finite signal constellations from lattices associated to rings of algebraic integers and their ideals.
		The procedure provides a natural way to label the constellation points by elements of a finite field.
		1 Lattice constellations and codes from quadratic number field..
	citeseer.ist.psu.edu /569926.html (397 words)

	Lattice (order)
		Consider an algebraic structure in the sense of universal algebra, given by (L,
		It now can be seen very easily that this operation really makes (L,
		), a homomorphisms of lattices is a function f : L → M with the properties that
	www.sciencedaily.com /encyclopedia/lattice__order_ (2523 words)

	Information and Computation Bibliography (Site not responding. Last check: 2007-10-08)
		A maximal monoidal closed category of distributive algebraic domains.
		In particular, \Pi(SUP), the full subcategory of BC with all prime-algebraic lattices as objects, is such a categorical semantics.
		PRIME is symmetric monoidal closed and maximal with respect to being closed under --\circ and \perp, if we demand that all objects are algebraic and distributive.
	theory.lcs.mit.edu /~iandc/References/huth1995:10.html (351 words)

	Information system
		Messages; carries a meaning to users or services.
		In the mathematical area of domain theory, a Scott information system (after its inventor Dana Scott) is a mathematical structure that provides an alternative representation of Scott domains and, as a special case, algebraic lattices.
		Facing it, always facing it, that's the way to get through.
	www.brainyencyclopedia.com /encyclopedia/i/in/information_system.html (268 words)

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