
	Where results make sense

Topic: Distributivity laws

Related Topics

Distributive lattice

Algebra over a field

In the News (Wed 25 Nov 09)

Northern Trust

Marvin Harrison

Peter Chernin

Gary Locke

Match Play

Penelope Cruz

Mickey Rourke

AR Rahman

Danny Boyle

Hugh Jackman

	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 (771 words)

	Distributivity -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-19)
		Distributivity is most commonly found in (Jewelry consisting of a circlet of precious metal (often set with jewels) worn on the finger) rings and (Click link for more info and facts about distributive lattice) distributive lattices.
		Each interpretation is responsible for different distributive laws in the Boolean algebra.
		Rings and distributive lattices are both special kinds of (Formation of masts, spars, sails, etc., on a vessel) rigs, certain generalisations of rings.
	www.absoluteastronomy.com /encyclopedia/d/di/distributivity.htm (904 words)

	Distributivity (order theory) - Wikipedia, the free encyclopedia
		In the mathematical area of order theory, there are various notions of the common concept of distributivity, applied to the formation of suprema and infima.
		There is still more to say about complete distributivity and its intuitionistic variants: see the article on completely distributive lattices.
		Distributivity is a basic concept that is treated in any textbook on lattice and order theory.
	en.wikipedia.org /wiki/Distributivity_(order_theory) (766 words)

	All words on Distributivity (Site not responding. Last check: 2007-10-19)
		# 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.
		# Logical disjunction ("or") is distributive over logical conjunction ("and"), and conjunction is distributive over disjunction.
		# For integers, the greatest common divisor is distributive over the least common multiple, and vice versa: gcd(''a'',lcm(''b'',''c'')) = lcm(gcd(''a'',''b''),gcd(''a'',''c'')) and lcm(''a'',gcd(''b'',''c'')) = gcd(lcm(''a'',''b''),lcm(''a'',''c'')).
	www.allwords.org /di/distributivity.html (929 words)

	Distributivity - Encyclopedia Glossary Meaning Explanation Distributivity (Site not responding. Last check: 2007-10-19)
		Here you will find more informations about Distributivity.
		* * is distributive over + if it is both left- and right-distributive.
		# For real numbers, addition distributes over the maximum operation, and also over the minimum operation: a + max(b,c) = max(a+b,a+c) and a + min(b,c) = min(a+b,a+c).
	www.encyclopedia-glossary.com /en/Distributivity.html (773 words)

	meet
		The laws of absorption ensure that both definitions are indeed equivalent.
		Since any lattice comes with two binary operations, it is natural to consider distributivity laws among them.
		An overview of these different notions is given in the article on distributivity in order theory.
	www.cooldictionary.com /words/meet.wikipedia (2315 words)

	Distributive lattice (Site not responding. Last check: 2007-10-19)
		Probably the most common type of distributivity is the one defined for lattices, where the formation of binary suprema and infima provide the total operations of join(
		For a complete lattice, arbitrarysubsets have both infima and suprema and thus infinitary meet and join operations are available.
		There is still more to sayabout complete distributivity and its intuitionistic variants: see the article on completelydistributive lattices.
	www.therfcc.org /distributive-lattice-84917.html (717 words)

	iqexpand.com (Site not responding. Last check: 2007-10-19)
		Also, conjunction is distributive over exclusive disjunction (andquot;xorandquot;).
		From comp.compilers newsgroup: Distributivity and types Distributivity and types andquot;Sanjay Pujareandquot; andlt;sanjayp@ddi.comandgt; 22 May 1999 03:01:25 -0400 From comp.compilers Related articles Distributivity...
		Distributivity Next: Complementary set Up: Operations with sets Previous: Union Contents Distributivity Proposition 3.2.18 (distributive law) Let, and be three sets.
	distributivity.iqexpand.com /index.php?title=Near-rig&action=edit (910 words)

	CS 540 - Introduction to Artificial Intelligence (Site not responding. Last check: 2007-10-19)
		Commutivity Laws All binary connectives are commutative EXCEPT the conditional → connective.
		p ∧ q ⇔ q ∧ p p ∨ q ⇔ q ∨ p p ∧ q ↔ q ∧ p p ⊕ q ⇔ q ⊕ p Associativity Laws All binary connectives are associative EXCEPT the conditional → connective.
		But once you have brought the quantifiers out, then you have imposed dependencies over the variables, and you then you must pay strict attention to which quantifiers you swap.
	www.cs.wisc.edu /~mschultz/cs540/equivalencelaws.html (99 words)

	Order theory glossary (Site not responding. Last check: 2007-10-19)
		A Boolean algebra is a distributive lattice withleast element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ^¬x = 0 and x v ¬x = 1.
		In other cases, complete (meet-)semilattices are defined to be bounded complete cpos, which is arguably the most complete class of posets thatare not already complete lattices.
		A meet- semilattice is distributive if for allelements a, b and x, a ^ b ≤ x implies the existence of elementsa' ≥ a and b' ≥ b such that a' ^ b' = x.
	www.therfcc.org /order-theory-glossary-210683.html (2502 words)

	Lattice (order) biography .ms (Site not responding. Last check: 2007-10-19)
		Furthermore, it turns out that the idempotency laws can be deduced from absorption and thus need not be stated separately.
		The above laws for absorption ensure that both definitions are indeed equivalent.
		For example, the Knaster-Tarski theorem states that the set of fixed points of a monotone function on a complete lattice is again a complete lattice.
	meet.biography.ms (2733 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		All the $2^n$-ons have $1$ and obey numerous identities including weakened distributive and associative laws.
		In the case of 16-ons these weakened distributivity laws characterize them, i.e.
		distributive) over the reals and complex numbers are impossible in dimensions other than powers of 2.
	dimacs.rutgers.edu /Events/2002/abstracts/smith.html (266 words)

	Encyclopedia: Distributivity (Site not responding. Last check: 2007-10-19)
		Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles.
		In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite lineâ€”the number line.
		In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers.
	www.nationmaster.com /encyclopedia/Distributivity (795 words)

	Algebra Boolean Distributivity (Site not responding. Last check: 2007-10-19)
		Here are proofs of distributivity, modularity, CC, and B1 from the BA 4-basis...
		correspond to equivalences of boolean algebra: distributivity, commutativity, identity, and excluded middle...
		mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that...
	www.online-math.com /3/algebra-boolean-distributivity.html (261 words)

	Matrices - Introducing Mathematics - Page 7 (Site not responding. Last check: 2007-10-19)
		Although in general matrix multiplication does not obey the commutivity law, it does always obey the associativity law, that is, if A, B and C are matrices that can be multiplied together (i.e.
		The matrices also satisfy A.(B + C) = A.B + A.C and (A + B).C = A.C + B.C which are known as the distributivity laws.
		This matrix has 1's going down the diagonal from the top left down to the bottom right; the letter n denotes how many rows and columns it has (i.e.
	introducingmathematics.com /matrices/07.html (122 words)

	3. A Strict Computation Model for
		Thus it makes a difference whether distributivity laws for pushing a complement into expressions have been applied.
		The normal form is justified by the de-Morgan laws of the calculus, as well as the several distributivity laws for translation.
		A lot of further partial evaluation which follows from the equational laws could be applied on normalization; however, apart of complement pushing, none of these really affects the power of the computation model.
	www.cs.unt.edu /~idl99/Proceedings/grieskamp/mz/node3.html (1482 words)

	emet information,meet (Site not responding. Last check: 2007-10-19)
		) constitute two semilattices, while the absorption laws guarantee that both of these structures interact appropriately.Furthermore, it turns out that the idempotency laws can be deduced from absorption and thus need not be stated separately.
		For example, the Knaster-Tarski theorem states that the set of fixed points of a monotone function on a complete lattice is again a completelattice.
		An overview of these different notions is given in thearticle on distributivity in ordertheory.
	www.vsearchmedia.com /emet.html (2349 words)

	[No title]
		In Mizar, as yet, the system of semantic correlates is close to abstract syntax, but: All propositional connectives but $\land$, $\neg$ and $\top$, are eliminated using well known rules (de Morgan laws etc.), the only explanation needed: the equivalence $a \iff b$ is equal to $$(a \to b) \land (b \to a)$$ i.e.
		I think that it is enough for the beginning, just observe that some of logical laws are obvious, because we work with semantic correlates.
		We experimented earlier with the canonical normal form (all disjuncts of the same length) it gives a bit stronger concept of obviousness, and the regular normal form (it is element of a Heyting algebra), but it complicates things and the gain is comparatively small.
	www.cs.ru.nl /~freek/mizar/by.tex (2696 words)

	[CoFI] Properties of relations
		Boolean algebra Premiss: informally, albeit precisely in a mathematical sense, the concept of Boolean algebra can be succinctly defined as that of a "bounded complemented distributive lattice".
		Before recalling the traditional solution (Huntington), and its (recently proven equivalent) Robbins' "dual" version, here's a third option, which requires eight equations, but somewhat simpler ones: commutativity of the two binary operations, the two distributivity laws, bound definitions (1 = x + -x, 0 = x.
		In CASL terms, it would still be sensible to put the "assoc, comm, unit..." properties as they stand now, so four more equations would do: distributivity and bound definitions.
	mailman.daimi.au.dk /pipermail/cofi-discuss/2003q4/000047.html (481 words)

	CONK! Encyclopedia: Order_theory (Site not responding. Last check: 2007-10-19)
		This condition is called distributivity and gives rise to distributive lattices.
		There are some other important distributivity laws which are discussed in the article on distributivity in order theory.
		Some additional order structures that are often specified via algebraic operations and defining identities are
	www.conk.com /search/encyclopedia.cgi?q=Order_theory (4020 words)

	[No title]
		ring +------------------------------------------------------------ A ring (X,+,,0) is a set X with a binary operation + and a binary operation such that (X,+,0) is a commutative group and (X,) is a semigroup and such that the distributivity* laws a(b+c) = ab + ac, (a+b)c - ac+bc hold.
		Examples: the integers Z form a ring with addition and multiplication the set of rational numbers Q, the set of real numbers R or the complext numbers C form a ring with addition and multiplication.
		semigroup +------------------------------------------------------------ A semigroup (X,+) is a set X with a binary operation + which satisfies the associativity law (a+b)+c = a+(b+c).
	www.math.harvard.edu /~knill/sofia/data/algebra.txt (1599 words)

	Kids Be Safe : Article 'Glossary of order theory' (Site not responding. Last check: 2007-10-19)
		A Boolean algebra is a distributive lattice with least element 0 and greatest element 1, in which every element x has a complement ¬x, such that x ^ ¬x = 0 and x v ¬x = 1.
		A lattice L is called distributive if, for all x, y, and z in L, we find that x ^ (y v z) = (x ^ y) v (x ^ z).
		A meet-semilattice is distributive if for all elements a, b and x, a ^ b ≤ x implies the existence of elements a' ≥ a and b' ≥ b such that a' ^ b' = x.
	www.kidsbesafe.org /DisplayArticleFull373235.html (6901 words)

	Citations: Functorial operational semantics and its denotational dual - Turi (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		We then insist on the condition that 7 is a distributivity law for comonads (the name is by analogy with distributivity laws for monads as in [24] they originated in
		The role of such distributivity laws will be apparent from the proof of Proposition 69.
		This appeal to distributivity law subsumes the above, simple case where preserves the comonad [ just take 7c to be the identity.
	citeseer.ist.psu.edu /context/222314/0 (2592 words)

	[No title]
		Suppose that a multiplication $k,x\mapsto k\odot x$ of elements of $V$ by elements of $K$ is given, and moreover, this multiplication is associative, distributes over the addition in $V$, and satisfies $\1\odot x= x$ for all $x\in V$.
		Hence it suffices to verify the distributivity laws and the associativity of the multiplication by elements of $K$.
		Since the opposite inequality is obvious, this completes the verification of the second distributivity law.
	sophus-lie.euro.ru /biblio/idempan/tensor.tex (6631 words)

	Lattice_(order) (Site not responding. Last check: 2007-10-19)
		Note that the laws for idempotency, commutativity, and associativity just state that (L, $\vee) and (L, \wedge) constitute two semilattices, while the absorption$ laws guarantee that both of these structures interact appropriately.
		One can now check that the relation ≤ introduced in this way defines a partial ordering within which binary meets and joins are given through the original operations $\vee and \wedge .$
		A lattice (L, $\vee, \wedge) is$ distributive, if the following condition is satisfied for every three elements x, y and z of L:
	www.freecaviar.com /search.php?title=Lattice_(order) (2466 words)

	[No title]
		A "rig" is like a ring, except one doesn't ask for additive inverses (joke: drop the "n"egatives); a "near-" thingummy is one in which one half of the distributivity laws doesn't hold.
		As for equational laws there are various implications of the weak distributivity of *,1
		They arise from the algebraic structure: the additive and multiplicative monoids, weak distributivity, etc. (The manipulations are reminiscent of those we need to bring ordinal notations into Cantor Normal Form.) Here is a datatype for arithmetical expressions.
	homepages.inf.ed.ac.uk /v1phanc1/arithmetic.lhs (5550 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		If a ring is commutative, then the two distributivity laws reduce down to the same thing, and the requirement that
		The distributive properties are the only properties in the definition of a ring that involve both operations.
		There is a cancellation law of multiplication for the ordinary number systems.
	www.math.nmsu.edu /~pmorandi/math331s01/BookHandoutPart4.html (2064 words)

	Citations: A structure-preserving clause form translation - Plaisted, Greenbaum (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		....as happening in the translation based on distributivity laws using the technique of Lifschitz et al.
		The size of the converted formula can be exponential with respect to the size of f, the worst case occurring when f is in disjunctive normal form.
		Distributed Problem-Solving as Concurrent Theorem Proving - Fisher, Wooldridge (1997)
	citeseer.ist.psu.edu /context/22916/0 (1077 words)

	Citations: Kripke models for linear logic - Allwein, Dunn (ResearchIndex) (Site not responding. Last check: 2007-10-19)
		EXPLICIT NEGATIONS 479 For the logics with distribution, we can give simple semantics for # and # as set intersection and union respectively.
		Allwein and Hartonas [3] report the full duality theorem for lattices, extending the duality to one for congruences and sublattices, and obtaining a sheaf representation of lattices.
		The present paper may be seen as an application of an extension of Dunn s representation of galois connections in [18] and is part of the first author s....
	sherry.ifi.unizh.ch /context/19432/0 (1964 words)

	NTU Info Centre: Lattice (order) (Site not responding. Last check: 2007-10-19)
		Note that the laws for idempotency, commutativity, and associativity just state that (L,) and (L,) constitute two semilattices, while the absorption laws guarantee that both of these structures interact appropriately.
		A lattice (L,,) is distributive, if the following condition is satisfied for every three elements x, y and z of L:
		A strictly weaker property is modularity: a lattice (L,,) is modular if, for all elements x, y, and z of L, we have
	www.nowtryus.com /article:Lattice_(order) (2757 words)

	IWFMS'04 --- Invited Talks (Site not responding. Last check: 2007-10-19)
		We propose a natural notion of logical relations able to deal with the monadic types of Moggi's computational lambda-calculus.
		The treatment is categorical, and is based on notions of subsconing and distributivity laws for monads.
		Our approach has a number of interesting applications, including cases for lambda-calculi with non-determinism (where being in logical relation means being bisimilar), dynamic name creation, and probabilistic systems.
	www.lsv.ens-cachan.fr /~zhang/workshop/abstracts/nowak.html (84 words)

Try your search on: Qwika (all wikis)