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# Topic: Greatest element

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 PlanetMath: maximal element ) have no least element, but infinitely many minimal elements (the primes.) In neither case is there a greatest or maximal element. have no least element or minimal element, and no maximal or greatest element. This is version 6 of maximal element, born on 2002-03-02, modified 2006-10-28. planetmath.org /encyclopedia/MinimalElement.html   (111 words)

 Greatest element - Wikipedia, the free encyclopedia Greatest elements of a partially ordered subset must not be confused with maximal elements of such a set. The least and greatest elements of the whole partially ordered set play a special role and are also called bottom and top or zero (0) and unit (1), respectively. The existence of least and greatest elements is a special completeness property of a partial order. en.wikipedia.org /wiki/Top_element   (345 words)

 Greatest element - Wikipedia, the free encyclopedia In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. Hence, the greatest element of S is an upper bound of S that is contained within this subset. This also demonstrates that the existence of a least upper bound (the number 1 in this case) does not imply the existence of a greatest element either. en.wikipedia.org /wiki/Greatest_element   (345 words)

 Greatest element -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-16) Hence, the greatest element of S is an ((mathematics) a number equal to or greater than any other number in a given set) upper bound of S that is contained within this subset. Greatest elements of a partially ordered subset must not be confused with (Click link for more info and facts about maximal element) maximal elements of such a set. The existence of least and greatest elements is a special (Click link for more info and facts about completeness property) completeness property of a partial order. www.absoluteastronomy.com /encyclopedia/g/gr/greatest_element.htm   (379 words)

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 meet   (Site not responding. Last check: 2007-10-16) The greatest element is often denoted by 1 and the least element by 0. The supremum is given by the least common multiple and the infimum by the greatest common divisor. A prime filter is a filter F with the additional property that for all elements a and b in L, if avb in F then either a in F or b in F. www.yourencyclopedia.net /meet.html   (1519 words)

 Ordinal number - LearnThis.Info Enclyclopedia   (Site not responding. Last check: 2007-10-16) For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}. Whenever you have two ordinals S and T, S is an element of T if and only if S is a subset of T, and moreover, either S is an element of T, or T is an element of S, or they are equal. An ordinal is finite if and only if the opposite order is also well-ordered, which is the case if and only if each of its subsets has a greatest element. encyclopedia.learnthis.info /o/or/ordinal_number.html   (1357 words)

 Bounded complete - Wikipedia, the free encyclopedia Note also that the term bounded poset is sometimes used to refer to a partially ordered set which has both a least and a greatest element. Hence it is important to distinguish between a bounded complete poset and a bounded cpo. Now these elements can be ordered based on the prefix order of words: a decimal number n is below some other number m if there is some string of digits w such that nw = m. en.wikipedia.org /wiki/Bounded_complete   (441 words)

 Compact element -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-16) Compact elements are important in (Click link for more info and facts about domain theory) domain theory, where they are considered as a kind of primitive element: the information represented by compact elements cannot be obtained by any approximation that does not already contain this knowledge. On the other hand, it may happen that all non-compact elements can be obtained as directed suprema of compact elements. It may well be that this is the only compact element, as the example of the (An old small silver Spanish coin) real unit interval [0,1] shows. www.absoluteastronomy.com /encyclopedia/C/Co/Compact_element.htm   (617 words)

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 Animation: Heapsort   (Site not responding. Last check: 2007-10-16) The sorting algorithm Heapsort is based on the data structure where the greatest element is the root of the binary tree. Remove the root of the heap – the greatest element of the partial tree – and swap it with the last leave (e.g. the element with the greatest index) Following that the binary tree has to be converted to a heap again as the new root probably is not greater than its children and so does not conform to the heap specification. www.informatik.uni-siegen.de /db/singleAnim.php3?lang=en&anim=heapsort.en   (281 words)

 Read about Maximal element at WorldVillage Encyclopedia. Research Maximal element and learn about Maximal element here!   (Site not responding. Last check: 2007-10-16) The definition for minimal elements is obtained by using ≥ instead of ≤. This example also shows that maximal elements are usually not unique and that it is well possible for an element to be both maximal and minimal at the same time. It is easy to see that any maximal element of such a subset must be the unique greatest element. encyclopedia.worldvillage.com /s/b/Maximal_element   (399 words)

 Order theory Details, Meaning Order theory Article and Explanation Guide These are graphss where the vertices are the elements of the poset and the ordering relation is indicated by both the edges and the relative positioning of the vertices. For example, 0 is the least element of the natural numbers and the empty set is the least set under the subset order. Greatest lower bounds in turn are given by the greatest common divisor. www.e-paranoids.com /o/or/order_theory.html   (4039 words)

 Maximal element   (Site not responding. Last check: 2007-10-16) In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S. What is important to note about maximal elements is that they are in general not the greatest elements of a subset S, i.e. Yet, in a totally ordered set, the terms maximal element and greatest element coincide, which is why both terms are used interchangeably in fields like analysis where only total orders are considered. www.sciencedaily.com /encyclopedia/maximal_element   (450 words)

 Science Fair Projects - Scott domain   (Site not responding. Last check: 2007-10-16) every element of D can be obtained as the supremum of a directed set of compact elements of D. Also note that, while the term "Scott domain" is widely used with this definition, the term "domain" does not have such a general meaning: it may be used to refer to many structures in domain theory and is usually explained before it is used. The empty word is the bottom element of this ordering and every directed set (which is always a chain) is easily seen to have a supremum. www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Scott_domain   (711 words)

 Greatest lower bound - Computing Reference - eLook.org The greatest lower bound of a set S is the greatest element b such that for all s in S, b <= s. The glb of mutually comparable elements is their minimum but in the presence of incomparable elements, if the glb exists, it will be some other element less than all of them. (In LaTeX "<=" is written as \sqsubseteq, the glb of two elements a and b is written as a \sqcap b and the glb of set S as \bigsqcap S). www.elook.org /computing/greatest-lower-bound.htm   (140 words)

 Read about Greatest element at WorldVillage Encyclopedia. Research Greatest element and learn about Greatest element ...   (Site not responding. Last check: 2007-10-16) partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. Greatest elements of a partially ordered subset must not be confused with The existence of least and greatest elements is a special encyclopedia.worldvillage.com /s/b/Bounded_poset   (347 words)

 BRUtil - Bounded/Realtime Utility Library: Interface SortedSet Returns the lowest element that is greater that the specified element. Returns the greatest element that is less that the specified element. the greatest element that is less that the specified element. www.cs.wustl.edu /~sjf1/brutil/brutil/SortedSet.html   (104 words)

 INFIMUM FACTS AND INFORMATION In mathematics the infimum of a subset of some set is the greatest_element, not necessarily in the subset, that is smaller than all other elements of the subset. Infima are in a precise sense dual to the concept of a supremum and thus additional information and examples are found in that article. In analysis the infimum or greatest lower bound of a set ''S'' of real_numbers is denoted by inf(''S'') and is defined to be the biggest real number that is smaller than or equal to every number in ''S''. www.whereintheworldisbush.com /infimum   (556 words)

 Lattice_(order) 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. It is characterized as being greatest among all elements y with the property that x $\wedgey = 0.$ 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. www.freecaviar.com /search.php?title=Lattice_(order)   (2466 words)

 Analysis WebNotes: Chapter 02, Class 05 If a set has a greatest element then that greatest element is the supremum of the set. On the other hand a set may have no greatest element and still have a supremum (for example, this set studied in Class 4). So, provided X has no smallest element (which is typically the case), the empty set cannot be relied on to have a supremum. www.math.unl.edu /~webnotes/classes/class05/class05.htm   (373 words)

 Course Summary Reflexive property: (a,a) is an element of R for every a in A. Symmetric property: if (a,b) is an element of R, (b,a) is an element of R. Antisymmetric property: if (a,b) is an element of R and (b,a) is an element of R, then a=b. An element a is the greatest element if b<=a for all b in S. An element a is the least element if a<=b for all b in S. Greatest and least elements are unique when they exist. An element that is greater than all the elements in a subset A of a poset is called an upper bound of A. sweb.uky.edu /~jcscov0/dmath_4.htm   (1397 words)

 The Basic Theory of Ordering Relations: A Supplement to Quantum Logic and Probability Theory The set of subsets of a finite set having an even number of elements is an example of a poset that is not a lattice. A complement for an element p of a (bounded) lattice L is another element q such that p Elements in the range of cl are said to be closed; those in the range of int are said to be open. plato.stanford.edu /entries/qt-quantlog/supplement2.html   (1398 words)

 Physics Help and Math Help - Physics Forums - Tension paradox? When the element of the string has displacement zero, it has the greatest possible kinetic energy and elastic potential energy, and is stretched by the greatest amount possible. When the element of the string has displacement equal to the amplitude of the wave, it has zero kinetic energy, zero elastic potential energy, and is stretched by the smallest amount possible. However, when the element is rushing through its zero displacement position, it is stretched to its maximum extent, and its elastic potential energy then is a maximum. www.physicsforums.com /printthread.php?t=35610   (902 words)

 Order Relation   (Site not responding. Last check: 2007-10-16) That is, every element is related with every element one way or the other. The elements in a finite poset can be ordered linearly in a number of ways while preserving the partial order. The basic idea of the topological sorting is to first remove a minimal element from the given poset, and then repeat that for the resulting set until no more elements are left. www.cs.odu.edu /~toida/nerzic/content/relation/order/order.html   (1120 words)

 Aluminum   (Site not responding. Last check: 2007-10-16) Shortly thereafter, the name aluminium was adopted to conform with the "ium" ending of most elements, and this spelling is now in use elsewhere in the world. Pure aluminum is soft and lacks strength, but it can be alloyed with small amounts of copper, magnesium, silicon, manganese, and other elements to impart a variety of useful properties. The compounds of greatest importance are aluminum oxide, the sulfate, and the soluble sulfate with potassium (alum). www.scescape.net /~woods/elements/aluminum.html   (495 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-16) To prove (2), assume that x and y are elements of M. By the definition of M, this means that x and y belong to two ideals of Y, say x in K1 and y in K2. (2) A greatest element is an element M such that, for all x, x <= M. A greatest element is always a maximal element, but the converse is not true. For example, if you consider the set of integers > 1, and the relation a <= b iff b divides a, then any prime number is a maximal element (no other element divides it), but not a greatest element (it does not divide all the numbers under consideration). mathforum.org /library/drmath/view/65137.html   (954 words)

 NavigableSet Returns a view of the portion of this set whose elements are less than (or equal to, if inclusive is true) toElement. Returns the least element in this set strictly greater than the given element, or null if there is no such element. Returns a view of the portion of this set whose elements are greater than (or equal to, if inclusive is true) fromElement. java.sun.com /javase/6/jcp/mr2/apidiffs/java/util/NavigableSet.html   (1768 words)

 HeapSort Example Consider the values of the elements as priorities and build the heap tree. After performing step 2, the order of the elements will be opposite to the order in the heap tree. The element 10 is less than the children of the hole, and we percolate the hole down: www.simpson.edu /~sinapova/cmsc250/LN250_Weiss/L17-HeapSortEx.htm   (586 words)

 [No title]   (Site not responding. Last check: 2007-10-16) This is a lattice, since each element in the list divides the next one. The least upper bound of two numbers in the set is the larger, and the greatest lower bound is the smaller. This is a lattice - the least upper bound of two integers in the set is the smaller integer and the greatest lower bound of two integers in the set is the larger integer. www.dcs.gla.ac.uk /~pat/af2/tutorials/solutions/week12solns.doc   (381 words)

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