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Topic: Maximal element

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	PlanetMath: maximal subgroup
		For this reason, maximal subgroups are sometimes called maximal proper subgroups.
		maximal, maximal normal subgroup, maximal proper normal subgroup
		This is version 10 of maximal subgroup, born on 2002-02-19, modified 2007-06-13.
	planetmath.org /encyclopedia/Maximal2.html (105 words)

	Ruby Annotation
		The name of the enclosing element was chosen to compactly and clearly identify the function of the markup construct; the names for the other elements were chosen to keep the overall length short.
		element is an inline (or text-level) element that serves as an overall container.
		elements in the case of complex ruby markup.
	www.w3.org /TR/ruby (5124 words)

	NationMaster - Encyclopedia: Supremum (Site not responding. Last check: 2007-09-11)
		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. The term minimal element is defined dually.
		The term maximal element is also synonymous as long as one deals with real numbers or any other totally ordered set.
		For an example where there are no greatest but still some maximal elements, consider the set of all subsets of the set of natural numbers (the powerset).
	www.nationmaster.com /encyclopedia/supremum (3072 words)

	Extreme value - Biocrawler (Site not responding. Last check: 2007-09-11)
		Likewise, a greatest element of a poset is an upper bound of the set which is contained within the set, whereas a maximal element m of a poset A is an element of A such that if m ≤ b (for any b in A) then m = b.
		Any least element or greatest element of a poset will be unique, but a poset can have several minimal or maximal elements.
		In a totally ordered set, or chain, all elements are mutually comparable, so such a set can have at most one minimal element and at most one maximal element.
	www.biocrawler.com /encyclopedia/Maximum (336 words)

	Maximal element - TheBestLinks.com - Minimal element, Analysis, Directed set, Mathematics, ...
		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.
		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.
		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.thebestlinks.com /Minimal_element.html (442 words)

	NationMaster - Encyclopedia: Zorn's lemma (Site not responding. Last check: 2007-09-11)
		A maximal element of P is an element m ∈ P such that the only element x ∈ P with x ≥ m is x = m itself.
		If P is a poset in which every well-ordered subset has an upper bound, and if x is any element of P, then P has a maximal element that is greater than or equal to x.
		The Hausdorff maximal principle, (also called the Hausdorff maximality theorem) formulated and proved by Felix Hausdorff in 1914, is an alternate and earlier formulation of Zorns lemma and therefore also equivalent to the axiom of choice.
	www.nationmaster.com /encyclopedia/Zorn%27s-lemma (2012 words)

	zorn's lemma - Article and Reference from OnPedia.com
		A maximal element of P is an element m in P such that the only element x in P with m ≤ x is x = m itself.
		The ideal R was excluded because maximal ideals by definition are not equal to R.
		I is an ideal: if a and b are elements of I, then there exist two ideals J and K in T such that a is an element of J and b is an element of K.
	www.onpedia.com /encyclopedia/Zorn's-lemma (746 words)

	Suprema
		The term maximal element is also synonymous as long as one deals with real numbers or any other totally ordered set.
		The difference between the supremum of a set and the greatest element of a set may not be immediately obvious.
		For an example where there are no greatest but still some maximal elements, consider the set of all subsets of the set of natural numbers (the powerset).
	www.sfcrowsnest.com /scifinder/a/Suprema.php (1757 words)

	maximal_element - The Wordbook Encyclopedia
		It is clear that the only candidates to be maximal elements are those in the upper right arc of the circle.
		should be a greatest element or maximum but if fact it is not necessarily the case: the definition of maximal element is somewhat weaker.
		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.thewordbook.com /maximal_element (695 words)

	maximal ideal
		Consequently, a maximal ideal is a maximal element in the corresponding orders of proper ideals (i.e.
		Alternatively, maximal ideals are directly characterized to be those ideals which are subsets of only two ideals: the improper ideal and the maximal ideal itself.
		Ideals of order theory, including maximal ideals, are treated in ideal (order theory).
	www.abacci.com /wikipedia/topic.aspx?cur_title=maximal_ideal; (191 words)

	Mathematical Programming Glossary
		The problem is to maximize the flow from s to t subject to conservation of flow constraints at each node and flow bounds on each arc.
		In a partially ordered set, a maximal element is one for which no element follows it in the ordering.
		In particular, (0,1) is a maximal element, and its value is 1, which is not the maximum.
	glossary.computing.society.informs.org /index.php?page=M.html (2376 words)

	Kids.Net.Au - Encyclopedia > Kuratowski-Zorn lemma (Site not responding. Last check: 2007-09-11)
		A maximal element of P is an element m in P such that the only element x in P with m <= x is x = m itself.
		Like the well-ordering principle, Zorn's Lemma is equivalent to the axiom of choice, in the sense that either one together with the Zermelo-Fraenkel axioms of set theory is sufficient to prove the other.
		I is an ideal: if a and b are elements of I, then there exist two ideals J and K in T such that a ∈ J and b ∈ K.
	www.kids.net.au /encyclopedia-wiki/ku/Kuratowski-Zorn_lemma (823 words)

	Programming Taskbook 4.6 \| Minimums and maximums
		Find the minimal element and the maximal element of the sequence (that is, elements with the minimal value and the maximal value respectively).
		Find the amount of the elements that are located after the last maximal element.
		Find the amount of the elements that are located between the first and the last maximal element.
	sunschool.math.rsu.ru /pt4/minmax.htm (1128 words)

	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)

	Reference.com/Encyclopedia/Supremum
		In mathematics, given a subset S of an ordered set T, the supremum of S is the least element of T that is greater than or equal to each element of S.
		If S contains a greatest element, then that element is the supremum; and if not, then the supremum does not belong to the subset.
		In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S.
	www.reference.com /browse/wiki/Supremum (1674 words)

	Mathematical Programming Glossary
		The problem is to maximize the flow from s to t subject to conservation of flow constraints at each node and flow bounds on each arc.
		In a partially ordered set, a maximal element is one for which no element follows it in the ordering.
		In particular, (0,1) is a maximal element, and its value is 1, which is not the maximum.
	orion.math.uwaterloo.ca /~hwolkowi/mirror.d/glossary/index.php?page=M.html (2375 words)

	Zorn's lemma
		Note that u is an element of P but need not be an element of T.
		I is an ideal: if a and b are elements of I, then there exist two ideals J and K in T such that a ∈ J and b ∈ K.
		If P is a poset in which every well-ordered subset has an upper bound, and if x is any element of P, then P has a maximal element that is greater than or equal to x.
	www.ebroadcast.com.au /lookup/encyclopedia/zo/Zorn's_lemma.html (798 words)

	Ordered Sets : Software Foundations : Thomas Alspaugh : UCI
		Since it is the only maximal element, it is the maximum or top.
		The second set has three minimal elements, and an edge is drawn down from each of the three minimal elements of Q to each of the five maximal elements of P.
		The second set is an antichain, so all its elements are trivially minimal, and an edge is drawn down from each of the five minimal elements of P to the single maximal element of Q.
	www.ics.uci.edu /~alspaugh/foundations/orderedSet.html (2579 words)

	Zorn's lemma Summary
		An upper bound of a chain is an element which is greater than or equal to all elements in the chain.
		A maximal element of S in an element which is greater than or equal to all other elements of S. Zorn's lemma states if S is a partially ordered set with the property that every chain has an upper bound, then S has a maximal element.
		A maximal element of P is an element m ∈ P such that the only element x ∈ P with x ≥ m is x = m itself.
	www.bookrags.com /Zorn's_lemma (1382 words)

	Data.Set
		If the set already contains an element equal to the given value, it is replaced with the new value.
		Elements of the result come from the first set.
		Partition the set into two sets, one with all elements that satisfy the predicate and one with all elements that don't satisfy the predicate.
	haskell.org /ghc/docs/latest/html/libraries/base/Data-Set.html (422 words)

	An Ergodic Walk » maximum versus maximal
		In a partially ordered set S, an element x is a maximal element of S if there is no y in S such that x is less than y.
		An element x is a maximum of S if x is greater or equal than y for every y in S. If it exists, it is unique.
		A maximum (if there is one) is maximal, but there may be other maximal elements.
	www.ergodicity.net /?p=722 (432 words)

	Order Relation
		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)

	lec11Sept
		maximal if there is no element greater than it in the poset.
		the greatest element if it is greater than all other elements in the poset.
		the least element if it is less than all other elements in the poset.
	www.pitt.edu /~vanlehn/cs0441/lec30Oct.html (801 words)

	Other Jeu de Taquin and Hook Length Posets?
		None of the hundreds of small connected hook length posets have more than one maximal element, and so it is natural to conjecture that any connected hook length poset must have only one maximal element.
		It seems to be a coincidence that the number of 8 element posets with the hook length property is so close to the number of 8 element posets with the jeu de taquin property (232 versus 236, with 181 of each d-complete).
		Now construct a larger poset P as follows: Place a chain of n elements above Q, and let the minimal element of this chain cover each of the maximal elements of Q. It is easy to see that P will have the jeu de taquin property.
	www.math.unc.edu /Faculty/rap/AllAcctd.html (701 words)

	Math Forum: Ask Dr. Math: A Mathematical Essay
		Next, consider the set M of all proper filters that are generated by H. One of the elements in M, for example, could be the set of all cofinite sets, plus all even integers, plus all supersets and intersections.
		No element in M is principal, because since an element B of M contains all cofinite sets, B cannot contain one-element sets, for then B would contain the empty set (the intersection of one-element set and the cofinite complement), and this would contradict the fact that B is proper.
		Maximal, in this case, means that if we have any other set D in M which we can order according to the relation "F is a subset of D," then F's maximality implies that F and D are the same.
	mathforum.org /dr.math/faq/analysis_hyperreals.html (9036 words)

	Orðasafn: M
		maximal chain principle hákeðjulögmál Hausdorffs, = Hausdorff maximal principle.
		maximal tree spannandi tré, hátré, = scaffolding, = skeleton, = spanning tree.
		maximum element stærsta stak, efsta stak, hæsta stak, síðasta stak, = greatest element, = largest element, = last element, = maximum 2, -> max.
	www.hi.is /~mmh/ord/safn/safnM.html (2128 words)

	Maximal-Element Rationalizability
		While rationality formulated in terms of the choice of greatest elements according to a rationalizing relation has been analyzed relatively thoroughly in the earlier litera-ture, this is not the case for maximal-element rationalizability, except when it coincides with greatest-element rationalizability because of properties imposed on the rationalizing relation.
		We develop necessary and sufficient conditions for maximal-element rationaliz-ability by itself, and for maximal-element rationalizability in conjunction with additional properties of a rationalizing relation such as re exivity, completeness, P-acyclicity, quasi-transitivity, consistency and transitivity.
		"Maximisation and the Act of Choice," Papers 270, Banca Italia - Servizio di Studi.
	ideas.repec.org /p/mtl/montde/2002-16.html (560 words)

	BackgroundMaterial (Site not responding. Last check: 2007-09-11)
		A chain C is maximal if there is no chain C' containing C as a proper subset.
		Maximal and maximum antichains are defined in an analogous fashion.
		Without loss of generality, we may assume a_i is the minimal element of C_i and the maximal element of D_i for each i = 1, 2,...
	www.math.gatech.edu /~trotter/Section8-Posets.htm (1040 words)

	Working with Elements of a Braid Group
		Given elements u and v belonging to the same braid group B, return the left-gcd of u and v, that is, the with respect to preceq maximal element d of B satisfying d preceq u and d preceq v.
		Given elements u and v belonging to the same braid group B, return the right-gcd of u and v, that is, the with respect to succeq maximal element d of B satisfying u succeq d and v succeq d.
		Given a set or a sequence S containing elements of a braid group B, return the left-gcd of the elements of S, that is, the with respect to preceq maximal element d of B satisfying d preceq s for all s in S, where preceq is defined as above.
	www.math.lsu.edu /magma/text462.htm (6729 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: 2007-09-11)
		The largest and the smallest element of a set are called extreme values, absolute extrema, or extreme records.
		If an infinite chain S is bounded, then the closure Cl(S) of the set occasionally have a minimum and a maximum, in such case they are called the greatest lower bound and the least upper bound of the set S, respectivelly.
		Furthermore, if S is a subset of an ordered set T and m is the greatest element of S with respect to order induced by T, m is a least upper bound of S in T.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=maximum (382 words)

	Paper: Unknown :: Stanford University Libraries (Site not responding. Last check: 2007-09-11)
		For instance, 3 is the unique maximal element in a component (in P) of size m, is the maximum of all n elements in X then the probability that xi is clearly at least m/n, and it is also not difficult to show that is a non-maximal element
		J being the largest element in X are by Lemma 5.2.
		Suppose xi is the unique maximal element in a component c, a non-maximal eleme...
	computing.breinestorm.net /lemma+element+proof+father+maximal/5 (646 words)

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