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# Topic: Distributivity

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 PlanetMath: complete distributivity is a singleton, we have the concepts of join infinite distributivity and the meet infinite distributivity, intermediate between those of distributivity and complete distributivity. A lattice that satisfies both countable distributivity laws must be a countably complete lattice. A lattice that satisfies any of meet infinite distributivity, join infinite distributivity, meet continuity, or join continuity individually must be complete inherently. www.planetmath.org /encyclopedia/MeetInfiniteDistributivity.html   (333 words)

 Distributivity - Wikipedia, the free encyclopedia In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers. Distributivity is most commonly found in rings and distributive lattices. en.wikipedia.org /wiki/Distributivity   (781 words)

 Distributivity In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. Multiplication of numbers is distributive over addition of numbers, for a broad class of different kinds of numbers ranging from natural numbers to complex numbers and cardinal numbers. Distributivity is most commonly found in rings and distributive lattices. www.dejavu.org /cgi-bin/get.cgi?ver=93&url=http://articles.gourt.com/%22http%3A%2F%2Farticles.gourt.com%2F%3Farticle%3DDistributive   (795 words)

 Algebra Definitions Distributivity is a property of some sets and two binary operations, such as addition and multiplication. A set S and two operations form a field if three conditions are met: the set is a group under the first operation with commutativity, the set is a group under the second operation when the identity of the first operation is removed, and distributivity is satisfied. A set S and two operations form a ring if the set is a group under the first operation with commutativity, the set has associativity, closure, and identity under the second operation when the identity of the first operation is removed, and distributivity is satisfied. www.learner.org /channel/courses/learningmath/algebra/keyterms.html   (2598 words)

 PlanetMath: complete distributivity is a singleton, we have the concepts of join infinite distributivity and the meet infinite distributivity, intermediate between those of distributivity and complete distributivity. Cross-references: countable, infinite, meet, operation, join, distributivity, between, singleton, index, distributive lattice, subset, lattice This is version 3 of complete distributivity, born on 2006-02-18, modified 2006-03-16. planetmath.org /encyclopedia/CompleteDistributivity.html   (159 words)

 Fuzzy Logic: The Logic of Fuzzy Sets The distributivity properties also follow from properties of the maximum and minimum functions but the proof is a bit longer. The right-hand side of the first distributivity relation is (A ∩ B) ∪ (A ∩ C) which for fuzzy sets involves the evaluation of w = max(min(x,y),min(x,z)). The right-hand side of the second distributivity relation is (A ∪ B) ∩ (A ∪ C) which requires the evaluation of w = min(max(x,y),max(x,z)). www.sjsu.edu /faculty/watkins/fuzzysets.htm   (2246 words)

 Principle of distributivity - Wikipedia, the free encyclopedia The principle of distributivity states that the algebraic distributive law is valid for classical logic, where both logical conjunction and logical disjunction are distributive over each other. discusses the case that quantum logic is the correct logic, on the grounds that the principle of distributivity is inconsistent with a reasonable interpretation of quantum phenomena. For some logical propositions A, B and C, the principle of distributivity means that en.wikipedia.org /wiki/Principle_of_distributivity   (154 words)

 [No title] In all three cases, the distributivity of P over Q implies that of Q over P! For the lattices, these facts are well-known. Subject: re: distributivity Date: Tue, 30 Aug 1994 12:56:51 -0400 (EDT) From: Andreas Blass This is a comment tangential to Michael Barr's message about quantifiers. The pitfall here is that distributivity of binary sup over binary inf does not extend to distributivity of sups over infs of up-closed subobjects. www.mta.ca /~cat-dist/catlist/1999/distributivity   (969 words)

 [No title]   (Site not responding. Last check: 2007-10-27) This condition requires that the distributive operator long be used as long as all members of a domain are commented, that is, the domain is exhausted. The examples in (2) and (3) suggest that with or without explicit indication of distributivity, which is done by long ‘all’, universal quantifiers and plurals tend to have a distributive interpretation in Taiwanese. The NP distributed over by dou must be in the same local clause where dou is, and must be to dou’s left, though not necessarily immediately left. www.shakespeare.uk.net /journal/2_1/Wu2_1.doc   (15618 words)

 on polarities and linear logic The big problem with additives is distributivity which forces one to make dubious additive contractions. PtimesQ is presented as the closure of P.Q ; NparM is defined by duality PplusQ is presented as the closure of P+Q (the set of sums not the union!) ; observe that NwithM is not the intersection of N and M, although this holds when we restrict to positive parts. Such + are created by the plus rule (sum of contexts) or the structural rule of sum ; in both cases the two subnets are tied together by the focus, empty or not. www.seas.upenn.edu /~sweirich/types/archive/1991/msg00123.html   (1040 words)

 Weakly distributive categories   (Site not responding. Last check: 2007-10-27) In distributive and *-autonomous categories these operations can be regarded as, respectively, the and/or of traditional logic and the times/par of (multiplicative) linear logic. In the latter logic, however, the distributivity of product over sum is conspicuously absent: this paper studies a ``linearization'' of that distributivity which is present in both case. Furthermore, we show that this weak distributivity is precisely what is needed to model Gentzen's cut rule (in the absence of other structural rules) and can be strengthened in a natural way to generate *-autonomous categories. www.math.mcgill.ca /rags/linear/wdc.abstract.html   (265 words)

 Rutgers Linguistics: Christine Brisson   (Site not responding. Last check: 2007-10-27) The range of meanings of sentences with definite plurals is examined, and it is argued that a theory of distributivity that assigns universal force to distributed plural noun phrases is empirically inadequate. A change to the theory of distributivity is proposed in which the context-sensitive variable in the domain of the distributivity operator (the D operator) can be assigned a value that weakens its universal force. In addition, the limited distribution of all with collective predicates is accounted for by the proposal that lexically collective predicates belonging to certain aktionsart classes can contain a "hidden" D operator. equinox.rutgers.edu /people/alumni/abstract.php3?id=3   (323 words)

 Semantical considerations on linear logic notation Thus the distributivity criterion permits direct (categorical) sum to be paired with any of the tensor products but gives no criterion for preferring one tensor product as *the* mate of sum. Hence the calculus of binary relations is not a weakly distributive category, modified (1) and (2) constituting a weakened form of symmetry that does not hold for binary relations. In any event, unless a given tensor sum arises naturally as a useful operation in its own right, as opposed to merely one of several possible duals of a useful tensor product, it would seem to be rather a waste of time arguing about the proper notation for it. www.cis.upenn.edu /~bcpierce/types/archives/1991/msg00133.html   (1047 words)

 Jet Virtual Medium - Distributivity This allows for a distributed network topology composed of many servers communicating with each other in a dynamical manner, building up a potential global-wide medium of entities far away of each other able to use together the provided medium facilities. We refer here to distributed grouping as being the ability to send to all the entities that are part of a certain group in the whole medium. When a message is received by the multi-server distributed communication handler from a local connected entity, with the destination being a groupbroadcast inside a certain overall group, it simply sends the message to all the connected servers, leaving the task of actually sending to all the corresponding entities to the remote servers. www.cs.purdue.edu /homes/sion/jvm/docs/manual/3.htm   (1353 words)

 CIOS -   (Site not responding. Last check: 2007-10-27) He interviewed the pool of available women and eventually selected those with well-manicured fingernails, under the assumption that if they cared for their fingernails they would likely also care for the bombs. There are no doubt objective measures of morale, but we wouldn't want to overlook the expedient of asking the employees what the problem was and how things might be made better, which is in the realm of subjectivity. Their main job was observing a panel with 126 different gauges, instruments and displays and taking action upon the problems signaled. www.cios.org /mailboxes/Q-method/02054134.508   (1068 words)

 Li 1997, Dissertation Abstract   (Site not responding. Last check: 2007-10-27) It is argued in this thesis that the syntactic notion of tense is correlated with distributivity. If a language possesses tense markings, then a distributive reading of a preverbal NP is available even in the absence of an overt distributive marker. To derive a distributive reading of a preverbal NP in Chinese, an overt distributive element like dou must be used. aris.ss.uci.edu /ling/folder/abstracts/li_abstract.html   (294 words)

 Abstract of "Distributivity and Dependency"   (Site not responding. Last check: 2007-10-27) Sentences with multiple occurrences of plural definites give rise to certain effects suggesting that distributivity should be modeled by polyadic operations. In addition, it is shown that also the evidence against a simple atomic version of unary distributivity is not decisive. In the (uncommon) cases where distributivity with definites is not strictly atomic, they can be analyzed as dependent on implicit quantifiers. www.cs.technion.ac.il /~winter/papers/pludef.abs.html   (122 words)

 Citations: Tight Galois connections and complete distributivity - Raney (ResearchIndex) If L or M is completely distributive, then every Galois Connection [r; n] between L and M has a canonical.... Prove that a completely distributive lattice also satisfies the dual distributivity axiom: W i2I V A i = V f :I fi 0 [A i W i2I f(i) 2. If L is completely distributive then we obtain similarly that P L (X) is a supprojection of [O(X) L] and the way way below relation on P L (X) is induced by the way way below relation on [O(X) L] Let us also point out that for continuous X and completely distributive lattices L our.... citeseer.ist.psu.edu /context/25807/0   (1042 words)

 Median Algebra Proof (continued) After studying our first Otter proof of the long form of distributivity, Knuth sent us the following hand proof of the short form of distributivity. Here is how I think one can boil "median.pf" down to the shortest purely equational proof of the (short) distributive law that I know. Finally, we used this proof as hints to find a new proof of the long form of distributivity. www.cs.unm.edu /~veroff/MEDIAN_ALGEBRA/knuth_proof.html   (383 words)

 VAN GEENHOVEN (colloquium abstract)   (Site not responding. Last check: 2007-10-27) Distributive readings thus follow from the interaction of a pluractional verb with its temporal and nominal arguments; nonmaximality effects with a distributive predicate follow from the cumulative nature of the pluractional operator that is part of this predicate's meaning. As a result, we gain some refreshing insights in the close relationship between distributivity and cumulativity (see Landman, 1989) as well as between distributivity and the interpretation of temporal adverbial clauses that are standardly treated as restrictors in tripartite quantificational structures (see Krifka et al., 1995; Rothstein, 1995). This talk is the first part of a two-part presentation; the second will be in the spring quarter. ling.ucsc.edu /news_events/past_colloquia/abstracts/vangeenhoven.html   (414 words)

 Structural Operations You can use these primitives both to implement mathematical properties such as associativity and distributivity, and to provide the basis for some succinct and efficient programs. This distributes over lists, but does so only if the head of the whole expression is This distributes over lists, making sure that the head of the whole expression is documents.wolfram.com /v4/MainBook/2.2.10.html   (446 words)

 LINGUIST List 2.633: Yours, and Distributivity Re M. Kac's query concerning verbal morphological marking of distributivity: Languages that have this seem to be quite common, see the typological survey in: Xrakovskij, Viktor S. (ed.) 1989. If the example were "John carried two pianos upstairs", would you use the same collective/distributive distinction on the verb for the two possible interpretations (that he carried them both up at once, or that he took two trips). If the collective/distributive is spread over two different objects, any verb-based system would break down: "John and Bill carried two pianos upstairs" could be distributive on John and Bill and collective on the pianos, vice versa, or both collective, or both distributive. www.sfs.nphil.uni-tuebingen.de /linguist/issues/2/2-633.html   (536 words)

 Contiguity and distributivity in the enumerable Turing degrees   (Site not responding. Last check: 2007-10-27) Contiguity and distributivity in the enumerable Turing degrees We prove that a (recursively) enumerable degree is contiguous iff it is locally distributive. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility. www.math.wisc.edu /~lempp/papers/contig.html   (60 words)

 The CTK Exchange Forums At the bottom, there is a proof for the distributivity of multiplication over addition for natural numbers. The proof of the distributive law is by induction in z. Just before that derivation I say, "Let M be the set of all z for which the Law holds. www.cut-the-knot.org /htdocs/dcforum/DCForumID4/134.shtml   (257 words)

 LC '98 abstract: J. Leneutre   (Site not responding. Last check: 2007-10-27) Indeed it appears that full distributivity is supported by some Coherent spaces, considering some restrictions on their constructions. Starting from this remark, we aimed with DLL to build a ``distributive'' linear logic, which sequent calculus mimics these restrictions on Coherent spaces, thus staying ``linear''. A drawback lies in the fact that supplementary distributivity is only available for some formulas making the notion of provability of DLL non stable over substitution of formulas. www.math.cas.cz /~lc98/abstracts/Leneutre.html   (351 words)

 Two Universal Quantifiers in Taiwanese: Collectivity, Distributivity, and Scopes   (Site not responding. Last check: 2007-10-27) The Locality Condition restricts the distributive ability of the operator in the local clause where the operator is. Though plurals are similar to universal quantifiers with respect to distributivity in that both of them allow for a distributive interpretation, they are different with respect with the interaction with negation. In (43b), Q is instantiated to x, which is bound by a universal quantifier and hence the constraint on Q is met. www.shakespeare.uk.net /journal/2_1/wu2_1.html   (9000 words)

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