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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)

PlanetMath: relative complement A complement of an element in a lattice is only defined when the lattice in question is bounded. Conversely, a complemented lattice is relatively complemented if it is modular. This is version 6 of relative complement, born on 2006-04-21, modified 2006-04-22. planetmath.org /encyclopedia/RelativeComplement.html   (185 words)

PlanetMath: complemented lattice A complemented lattice is a bounded lattice in which every element has a complement. In a complemented lattice, there may be more than one complement corresponding to each element. This is version 12 of complemented lattice, born on 2005-02-16, modified 2006-06-19. planetmath.org /encyclopedia/PerspectiveElements.html   (178 words)

Complemented lattice - Wikipedia, the free encyclopedia In the mathematical discipline of order theory, and in particular, in lattice theory, a complemented lattice is a bounded lattice (that is it has a least element 0 and a greatest element 1), in which each element x has a complement, defined as an element y such that However in a distributive lattice, that is a lattice in which, for all x, y and z, the distributive law holds: Thus in a Boolean algebra, which is both a complemented distributive lattice and an orthocomplemented lattice, complements exist and are unique. en.wikipedia.org /wiki/Complemented_lattice   (177 words)

Springer Online Reference Works In an orthomodular lattice one studies distributivity, perspectivity, irreducibility, modularity of pairs, properties of the centre and of ideals, the commutator, solvability, and applications in the logic of quantum mechanics (see [1], [2]). Orthomodular lattices, which are a natural generalization of lattices of projections of factors, also constitute an essentially broader class, in that many properties of lattices of projections are not valid for arbitrary orthomodular lattices. There exists an orthocomplemented modular lattice whose completion by sections is not orthomodular (whereas the completion by sections of a semi-modular orthocomplemented lattice is semi-modular and the lattice of projections of a von Neumann algebra is semi-modular). eom.springer.de /O/o070450.htm   (435 words)

pure   (Site not responding. Last check: 2007-10-11) Special elements in a lattice such as complements and relative complements allow for defining complemented lattices, relatively complemented lattices, Boolean lattices that are briefly discussed in Section 6 of this first chapter; 44 exercises close this section. A lattice L is semimodular if it satisfies the upper covering condition: for two elements or The semimodular lattices of finite length are charac-terized in Theorem 2, while Theorem 9 states that a lattice L of finite length is semimodular L is M-symmetric. A lattice L is said to be geometric if it is semimodular, it is algebraic, and the compact elements of L are exactly the finite joins of atoms of L. Theorem 5 states that every geometric lattice is isomorphic to a direct product of directly indecomposable geometric lattices. www.library.tuiasi.ro /ipm/vol15no14/pure.html   (2526 words)

\1cw 3 In this paper it is attempted to relate the definitions of the fuzzy subset lattices with the other classical definitions of Postean and Boolean lattices in the context of Universal algebra. §1 The λ-rainbow (or fibber) lattices and the λ-Fuzzy subsets lattices. In  conceiving  the idea of rainbow lattices I was partly influenced at least by the terminology, of the Kantian ideas of transcendental analytic and synthetic logic and of Aristotle’s term of “colors of the word”. www.softlab.ntua.gr /~kyritsis/PapersInMaths/LogicandFuzzy/TransFuz.htm   (1656 words)

Relatively complemented lattice - Wikipedia, the free encyclopedia In mathematics, a relatively complemented lattice is a lattice L in which for all a, b, c in L with a ≤ b ≤ c there is some x in L such that x ∨ b = c and x ∧ b = a. An element x with this property is a complement of b relative to the interval [a,c]. If the lattice is a Boolean algebra, then the complement of b relative to the interval [a, c] is a ∨ (~ b) ∧ c. en.wikipedia.org /wiki/Relatively_complemented_lattice   (175 words)

CMS Winter 2003 Meeting One of the major ways of obtaining finite lattices with a given congruence lattice is by using chopped lattices: a partial lattice obtained fom a finite lattice by chopping off the unit element. Our best result was that if we obtain a finite chopped lattice by taking two finite sectionally complemented lattices and identify the zeroes and an atom chosen from each, then the resulting sectionally complemented chopped lattice has a sectionally complemented ideal lattice. With E. Schmidt, in 1962, from a finite poset we constructed a sectionally complemented chopped lattice and we proved that the ideal lattice of this chopped lattice is again sectionally complemented. www.cms.math.ca /Events/winter03/abs/ua.html   (1710 words)

Mathematica Slovaca   (Site not responding. Last check: 2007-10-11) For a bounded lattice $K$ and an element $a$ of $K - \{0,1\}$, we directly describe the structure of the lattice freely generated by $K$ and an element $u$ subject to the requirement that $u$ be a complement of $a$. As an application, we give a short and direct proof of the classical result of R.~P.~Dilworth (1945): {\it Every lattice can be embedded into a uniquely complemented lattice}. GRÄTZER, G.—LAKSER, H.: Freely adjoining a complement to a lattice, Math. www.mat.savba.sk /maslo/paper.php?id_paper=651   (124 words)

George Gratzer - On the endomorphism monoids of (uniquely) complemented lattices Recall that uniquely complemented lattices are very difficult to construct. Dilworth in 1945 solved a long standing conjecture of lattice theory by proving that not every uniquely complemented lattice is distributive (Boolean). He proved this by examining free lattices with a ``free'' complement operation. www.cms.math.ca /Events/winter99/abstracts/node151.html   (197 words)

AMCA: Finite lattices and congruences: The good, the bad, and the ugly by George Gratzer   (Site not responding. Last check: 2007-10-11) Lattice congruences are not as nice as congruences of groups and rings: There is no single class, in general, determining a congruence. If being ``nice'' is an algebraic property such as being semimodular or sectionally complemented, then we have tried in many instances to prove a much stronger form of these results by verifying that every finite lattice has a congruence-preserving extension that is ``nice''. We shall conclude with some recent results on the spectrum of a congruence of a finite, sectionally complemented lattice, measuring the sizes of the congruence classes. at.yorku.ca /c/a/i/g/63.htm   (312 words)

Margaret Masterman's "Theism as a Scientific Hypothesis"   (Site not responding. Last check: 2007-10-11) Considered as a lattice, the system which we have just constructed is the very well-known Boolean lattice of three minimals, the "cube" lattice. It is not the only finite Boolean lattice; the Boolean lattices go up in size, within lattice-theory, by having the numbers of their elements correspond to the sequence of the powers of two: i.e. It is a modular and a distributive lattice (modularity and distributiveness both being properties which define differing types of regularity); it is a self-dual lattice (i.e. home.earthlink.net /~hipbone/IDTWeb/Master.html   (2635 words)

NSDL Metadata Record -- orthocomplemented lattice orthocomplemented lattice is a complemented lattice in which every element has a distinguished complement, called an orthocomplement, that behaves like the complementary subspace of a subspace in a vector space. Formally, let L be a complemented lattice and denote M the set of complements of elements of L. In addition to the example of the lattice of vector subspaces of a vector space cited above, let's look at the Hasse diagrams of the two finite complemented lattices below,... nsdl.org /mr/1658508   (214 words)

Mathematical Background The lattice of all subsets of some set is an example of a bounded distributive complemented lattice, for which all the identities hold. Lattices are especially important for representing ontology and the techiques for revising, refining, and sharing ontologies. A Leibniz-style of lattice is the ultimate refinement for a given set of attributes, and it may be useful when all possible combinations must be considered. www.jfsowa.com /logic/math.htm   (14436 words)

AMCA: Congruence-preserving extensions of lattices by E. Tamas Schmidt A lattice L is sectionally complemented if for every b >= a in L, there is an element c that is the complement of a in the interval [0, b]. Every \omega-congruence-finite lattice K has a \omega-congruence-finite, relatively complemented congruence-preserving extension L. Furthermore, if K has a zero, then L can be taken to have the same zero. Every lattice K such that Comp(K) is a lattice admits a congruence-preserving extension into a relatively complemented lattice. at.yorku.ca /c/a/g/l/09.htm   (510 words)

LATYPES The DISTRIBUTIVE LAW of LATTICES resembles the DISTRIBUTIVE LAW OF ARITHMETIC -- a X (b + c) = a X b + a X c -- except THAT THERE ARE TWO DISTRIBUTIVE LAWS IN LATTICE THEORY. Elsewhere, I've referred to a COMPLEMENTED DISTRIBUTIVE LATTICE as a "t-lattice". Note that, in the MODULAR LATTICE, MAX is the JOIN of 3 LATTICE terms. members.fortunecity.com /jonhays/lattice2.htm   (636 words)

MIT Combinatorics Seminar: Jonathan Farley   (Site not responding. Last check: 2007-10-11) If M is a finite complemented modular lattice with n atoms and D is a bounded distributive lattice, then the Priestley power M[D] is shown to be isomorphic to the poset of so-called normal elements of D^n, thus solving a problem of E. It is shown that there exist a finite modular lattice A not having M_4 as a sublattice and a finite modular lattice B such that A⊗B is not semimodular, thus refuting a conjecture of Quackenbush from 1985. It is shown that the tensor product of M_3 with a finite modular lattice B is supersolvable if and only if B is distributive, thus proving a conjecture of Quackenbush from 1985. www-math.mit.edu /~combin/archive/2004_fall/04_11_19_farley.html   (134 words)

Read This: Briefly Noted, December 2005 The first two results conspire to show that in a Boolean algebra prime ideals and maximal ideals coincide (a marvelous algebraic result in its own right) even as it is also featured in the derivation of the third fact, a topological representation theorem due to none other than M. Stone. Thus, a raison d’être for the study of lattices and ordered algebraic structures is the facility this subject imparts to demonstrating a certain class of results from algebra and topology. Lattices and Ordered Algebraic Structures is extensive and scholarly, dense but accessible. www.maa.org /reviews/brief_dec05.html   (2568 words)

Untitled Document   (Site not responding. Last check: 2007-10-11) This work shows how lattice theory can be used to develop quantitative measures of selected characteristics of knowledge structure representations, and how these measures can be used to assess individual persons' knowledge structure representations in a classroom setting. Under this description, the set of all possible knowledge structure representations for a given set of concepts are the elements of a complemented, distributive lattice ordered by set inclusion. Measures are developed to assess the dissimilarity between two knowledge structure representations, the local complexity of a concept in a knowledge structure, and the global complexity of a knowledge structure. cresst96.cse.ucla.edu /Summary/479young.htm   (161 words)

lattice - OneLook Dictionary Search Lattice, Lattice : A Glossary of Mathematical Terms [home, info] Phrases that include lattice: distributive lattice, lattice energy, complemented lattice, lattice constant, lattice homomorphism, more... Words similar to lattice: fretwork, grille, latticed, latticework, latticing, wicket, trellis, trelliswork, more... www.onelook.com /cgi-bin/cgiwrap/bware/dofind.cgi?word=lattice   (354 words)

Atlas: Huntington Varieties of Lattices by R. Padmanabhan   (Site not responding. Last check: 2007-10-11) In spite of these deep theorems, it is still hard to find “nice” and “natural” examples of uniquely complemented lattices that are not Boolean. The reason is that uniquely complemented lattices having a little extra structure most often turn out to be distributive. Similarly, a lattice variety K is said to be a Huntington Variety if every uniquely complemented member of K is distributive. atlas-conferences.com /cgi-bin/abstract/casq-04   (356 words)

PlanetMath: complemented lattice Cross-references: Boolean ring, distributive lattice, lattice, antichain, non-trivial element, chain, bounded lattice (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Complemented lattices, orthocomplemented lattices and posets) Lattices and power sets by Schneemann on 2006-02-18 14:08:10 planetmath.org /encyclopedia/Complement2.html   (178 words)

CMS Summer 2002 Meeting Let L be a lattice, and let a < b < c be elements of L. Dilworth: Every lattice can be embedded into a uniquely complemented lattice. A projective plane is a simple, complemented, modular lattice of height 3. www.cms.math.ca /Events/summer02/abs/ua.html   (753 words)

AARNEWS - August 2001   (Site not responding. Last check: 2007-10-11) The standard definition of Boolean algebra is that it is a uniquely complemented distributive lattice, but other steps along the way are of interest in the study of quantum logics. This was accomplished by supplying the axioms for a given type of lattice along with the negation of another axiom or set of axioms which are unique to the second type of lattice. The function values found in the candidate lattice were inserted into the input, forcing MACE to consider the same lattice but now with the axioms of an ortholattice included (5) and (6) in their original form (not negated). www-unix.mcs.anl.gov /AAR/issueaugust01/issueaugust01.html   (4635 words)

complemented - OneLook Dictionary Search Tip: Click on the first link on a line below to go directly to a page where "complemented" is defined. Example: "A fine wine is a perfect complement to the dinner" Phrases that include complemented: complemented lattice, relatively complemented, relatively complemented lattice, uniquely complemented www.onelook.com /?w=complemented   (187 words)

[No title] The lattice of convex $l$-subgroups of a lattice-ordered group. Martinez, J. Topological coordinatizations and dualities of Brouwer lattices. Proceedings of the University of Houston Lattice Theory Conference (Houston, Tex., 1973), pp. www.math.ufl.edu /fac/facmr/Martinez.html   (481 words)

FREECD I'll illustrate the RANK measure of a lattice by a simple case: 6-CDL, the complemented distributive lattice on factors of 6. (Two elements of a lattice are mutually complementary if their only "connections" are at the "bottom" & "top" of the lattice, as is the case here with 2 & 3. Elements of RANK 1 (2 & 3, above) are known as "atoms" of a lattice. members.fortunecity.com /jonhays/freecdrk.htm   (264 words)

posetlattice.html Note that this implies that any finite set of elements has a g.l.b. A complemented distributive lattice is called a Boolean algebra. The algebra of subsets of a given Set. www.umsl.edu /~siegel/SetTheoryandTopology/posetlattice.html   (108 words)

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