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Topic: Lexicographic order

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	PlanetMath: lexicographic order
		The lexicographic order yields a total order on the field of complex numbers.
		If the original set is well-ordered, the lexicographic ordering on the product is also a well-ordering.
		This is version 10 of lexicographic order, born on 2005-05-04, modified 2006-10-16.
	www.planetmath.org /encyclopedia/LexicographicOrder.html (114 words)

	Order theory - Wikipedia, the free encyclopedia
		Order theory is a branch of mathematics that studies various kinds of binary relations that capture the intuitive notion of a mathematical ordering.
		Orders appear everywhere - at least as far as mathematics and related areas, such as computer science, are concerned.
		These are graphs 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.
	en.wikipedia.org /wiki/Order_theory (4062 words)

	Lexicographical order - Wikipedia, the free encyclopedia
		An important exploitation of lexicographical ordering is expressed in the ISO 8601 date formatting scheme, which expresses a date as YYYY-MM-DD.
		This date ordering lends itself to straightforward computerized sorting of dates such that the sorting algorithm does not need to treat the numeric parts of the date string any differently from a string of non-numeric characters, and the dates will be sorted into chronological order.
		In algebra it is traditional to order terms in a polynomial, by ordering the monomials in the indeterminates.
	en.wikipedia.org /wiki/Lexicographic_order (495 words)

	Facts about topic: (Order theory) (Site not responding. Last check: 2007-10-21)
		Order theory is a branch of mathematic (additional info and facts about mathematic) s that studies various kinds of binary relation (additional info and facts about binary relation) s that capture the intuitive notion of a mathematical ordering.
		Orders appear everywhere - at least as far as mathematics and related areas, such as computer science (The branch of engineering science that studies (with the aid of computers) computable processes and structures), are concerned.
		The first order that one typically meets in primary school (A school for young children; usually the first 6 or 8 grades) mathematical education is the order ≤ of natural numbers (The number 1 and any other number obtained by adding 1 to it repeatedly).
	www.absoluteastronomy.com /encyclopedia/o/or/order_theory.htm (4861 words)

	Wordwizard Clubhouse - New meaning for [lexicographical]
		In the mathematical field of combinatorics ‘lexicographic ordering’ of the permutations of a set is defined such that the ordering of the permutations of the set {1,2,3} are 123, 132, 213, 231, 312, and 321 (which was easier to demonstrate by example than to express mathematically).
		In set theory the ‘lexicographic ordering’ rule is that two subsets are ordered by their smallest elements.
		And if you are familiar with matrices the lexicographic ordering of the elements of a 3x3 matrix ‘A’ will be A11, A12, A13, A21, A22, A23, A31, A32, A33 (the numbers are subscripts) with the letter identifying the matrix, the first number identifying the row, and the second number identifying the column.
	www.wordwizard.com /ch_forum/topic.asp?TOPIC_ID=19316 (938 words)

	Quiz 5 Notes
		In lexicographic order, the elements of A x B are (w,a), (w,b), (x,a), (x,b), (y,a), (y,b), (z,a), (z,b).
		In lexicographic order, the elements of B x A are (a,w), (a,x), (a,y), (a,z), (b,w), (b,x), (b,y), (b,z).
		This is lexicographic order on A x B x C based on numerical order in the first component and alphabetic order in each of the remaining two components.
	www-cse.ucsd.edu /groups/Gill-rawfiles/SolWeb_CSE20/u5notes.html (4251 words)

	LGL - Ordering of Characters and Strings
		Even when character ordering is logical, the usual filing of words in dictionaries does not correspond with the mathematical lexicographic ordering of strings, but some equivalence relation on characters is involved.
		Arranging characters, and by extension strings, in order of representation codes may be convenient and efficient when the effect is hidden in a program, but it is clearly not suitable for application programming.
		is a linear order on A. The value of function f may be interpreted as the rank of an element of A for the linear order.
	diwww.epfl.ch /researchlgl/ada/components/text_processing/ordering.html (3771 words)

	Twelf User's Guide - 7 Termination
		The termination orders we construct are lexicographic or simultaneous extensions of the subterm ordering explained in section 7.2 Subterm Ordering.
		All identifiers in the order specification of a termination declaration must be upper case, must occur in the call patterns, and no variable may be repeated.
		On first-order terms, that is, terms not containing lambda-abstraction, the subterm ordering is familiar: M < N if M is a strict subterm of N, that is, M is a subterm N and M is different from N.
	www.cs.cmu.edu /~twelf/guide-1-2/twelf_7.html (1218 words)

	1.3.4 Generating Permutations
		The most natural generation order is lexicographic, the order they would appear if they were sorted numerically.
		Lexicographic order for $n=3$ is $\{1,2,3\}$, $\{1,3,2\}$, $\{2,1,3\}$, $\{2,3,1\}$, $\{3,1,2\}$, and finally $\{3,2,1\}$.
		Although lexicographic order is aesthetically pleasing, there is often no particular reason to use it.
	www.cs.sunysb.edu /~algorith/files/generating-permutations.shtml (229 words)

	PlanetMath: hashing
		Of course, in order to be able to tell if a location in the hash table is occupied, we need a special value called a “tombstone” to delimit an empty spot.
		In order to guarantee that this will eventually get us to an empty space, hashing using this policy works best with a prime-sized hash table.
		This just means that the lexicographic order of the hashed object locations is the same as the lexicographic order of the keys:
	planetmath.org /encyclopedia/Hashing.html (1665 words)

	Conservative Pruning Rules
		We can also define a lexicographic ordering on the expressions based on the way they are derived.
		Among all the children of a node with mathematically equivalent expressions, choose the one that is smallest in the lexicographic ordering.
		The expression that is smallest in the lexicographic ordering is chosen because by rule 7, the subtree under such a node contains the subtree under a node representing the same expression and having the same parent node.
	www.scl.ameslab.gov /Publications/Gus/FiveMultiplications/node5.html (1803 words)

	Mathematica Documentation: Combinatorica
		These permutations are generated in minimum change order, where successive permutations differ by exactly one transposition.
		The number of vertices in a graph is termed the order of the graph.
		Since the divisibility relation is reflexive, transitive, and anti-symmetric, it is a partial order.
	documents.wolfram.com /mathematica/Add-onsLinks/StandardPackages/DiscreteMath/Combinatorica.html (2160 words)

	Permutations
		The initial permutation corresponds to the sorted order of these element Structure is the Container implementation used to maintain the permutation's local copy of s.
		Change the permutation to the first permutation in lexicographic order, the identity permutation.
		Change the permutation to the last permutation in lexicographic order, the reverse of the identity permutation.
	dimacs.rutgers.edu /~berryj/backup/manual/node94.html (316 words)

	Definitions
		The standard name for this is `graded reverse lexicographic order', and the emphasis is on the first two words.
		Note the double-negative from pure lexicographic order, but it is the grading from the sums of powers that turns out to be most significant.
		Lexicographic order gives the simplest answers, but it turns out to be (naturally) the hardest one to compute.
	www.apmaths.uwo.ca /~rcorless/AM563/NOTES/Mar_13_96/node6.html (376 words)

	RFC 3076 (rfc3076) - Canonical XML Version 1.0 (Site not responding. Last check: 2007-10-21)
		For UTF-16, the leading byte order mark is treated as an artifact of encoding and stripped from the UCS character data (subsequent zero width non-breaking spaces appearing within the UTF-16 data are not removed) [UTF-16, Section 3.2].
		Lexicographic comparison, which orders strings from least to greatest alphabetically, is based on the UCS codepoint values, which is equivalent to lexicographic ordering based on UTF-8.
		Also, a trailing #xA is rendered after the closing PI symbol for PI children of the root node with a lesser document order than the document element, and a leading #xA is rendered before the opening PI symbol of PI children of the root node with a greater document order than the document element.
	www.faqs.org /rfcs/rfc3076.html (5964 words)

	Linear Orders
		- order for the built-in numerical types, the lexicographic ordering for string, and for point the lexicographic ordering of the cartesian coordinates.
		Sometimes, a user may need additional linear orders on a data type T which are different from the order defined by compare.
		As a new feature all order based data types like dictionaries, priority queues, and sorted sequences offer a constructor which allows a user to set the internally used ordering at construction time.
	www-graphics.stanford.edu /courses/cs368/LEDA/node8.html (516 words)

	Crossword (Site not responding. Last check: 2007-10-21)
		You should write a program to construct the first, in lexicographic order, double linear crossword of length L for a given list of words.
		The words in the list are arranged in lexicographic order and no word is repeated.
		For the given input data set your program should write to the output file the first, in lexicographic order, double linear crossword with the given length.
	acm.uva.es /p/v7/768.html (402 words)

	Lexicographical order - Encyclopedia, History, Geography and Biography (Site not responding. Last check: 2007-10-21)
		Lexicographical order - Encyclopedia, History, Geography and Biography
		This encyclopedia, history, geography and biography article about Lexicographical order contains research on
		Lexicographical order, Case of multiple products, Monomials, See also and Order theory.
	www.arikah.net /encyclopedia/Lexicographic_order (592 words)

	Lexicographic order (Site not responding. Last check: 2007-10-21)
		In mathematics, the lexicographical order, ordictionary order, is a natural order structure of the cartesian product of two ordered sets.
		The name comes from its generalizing the order given to words in a dictionary : a sequence of letters (i.e.
		An important property of the lexicographical order is that it preserves well-orders, that is, if A and B are well-ordered sets, then the product set A ×B with the lexicographical order is also well-ordered.
	www.therfcc.org /lexicographic-order-43261.html (342 words)

	Lex Order: The simplest elimination order
		This order is useful for attempting to solve for
		If producing high order polynomials is a consequence of this fanaticism so be it.
		Once this order is set, one can use all of the commands in the preceeding section in exactly the same form.
	math.ucsd.edu /~ncalg/NCBIGDOC02/node265.html (261 words)

	The Monomial Order Package (Site not responding. Last check: 2007-10-21)
		Q with respect to inverse lexicographic order, NIL otherwise The second returned value is T if P=Q, otherwise it is NIL.
		ORDER and they should be term orders which are used on the first K and the remaining variables.
		ORDER when there is only one primary variable.
	alamos.math.arizona.edu /CGB/latex-doc/manual/node9.html (174 words)

	Linear Orders
		- order for the built-in numerical types, the lexicographic ordering for string, and for point the lexicographic ordering of the cartesian coordinates.
		Sometimes, a user may need additional linear orders on a data type T which are different from the order defined by compare.
		As a new feature all order based data types like dictionaries, priority queues, and sorted sequences offer a constructor which allows a user to set the internally used ordering at construction time.
	graphics.stanford.edu /courses/cs368/LEDA/node8.html (516 words)

	Citations: The rst-order theory of lexicographic path orderings is undecidable - Comon, Treinen (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		Only recently the decidability of the rst order theory of recursive path orders in the case of unary signatures has been proven [14] A signature is called unary if it consists of unary function symbols and constants.
		The result has to be contrasted with the undecidability results of the lexicographic path ordering [6] for the case of symbols with arity 2 and total precedence and for the case of unary signatures with partial precedence.
		We recall that lexicographic path ordering (lpo) and the recursive path ordering and many other orderings such as [13, 10] coincide in the unary case.
	citeseer.lcs.mit.edu /context/1439429/0 (784 words)

	Generating Subsets
		Lexicographic order - Lexicographic order is sorted order, and often the most natural order for generating combinatorial objects.
		Unfortunately, it is surprisingly difficult to generate subsets in lexicographic order.
		Gray Code - A particularly interesting and useful subset sequence is the minimum change order, wherein adjacent subsets differ by the insertion or deletion of exactly one element.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE152.HTM (1024 words)

	The Nephroid Lab: Gröbner Bases
		An example of such an order is the "lexicographic order" based on x > y > z.
		This ordering says that to compare two monomials m, m' we first see if one of them has smaller power of x and say this monomial is smaller in < if so.
		For a given monomial order <, any polynomial f will have a unique term whose monomial is largest in the < order, called the <-leading term of f (NB: we are ignoring the coefficient in front of each term to obtain a monomial, e.g.
	www.geom.uiuc.edu /~fjw/calc-init/nephroid/grobner.html (790 words)

	CSCE 235 Discrete Math Project-Generating Permutations and Combinations
		Often, the permutation of a set will be given in lexicographic ordering.
		in lexicographic order is given for a set {1,2,3,...,n}.
		Then place the remaining three integers in lexicographic order to obtain the permutation 364125.
	www.nebraskaroads.com /csce235/section4_7.html (613 words)

	Twelf User's Guide - 8 Termination
		The termination orders we construct are lexicographic or simultaneous extensions of the subterm ordering explained in section 8.3 Subterm Ordering.
		order = order % equal rdecl ::= pdecl callpats % reduction predicate decl ::=...
		In order to verify that the definition of gcd terminates, we need to show that the arguments in the recursive call are decreasing, i.e.
	www.cs.cmu.edu /People/twelf/guide/twelf_8.html (1616 words)

	Reverse lexicographic order (Site not responding. Last check: 2007-10-21)
		The most important monomial order is the (graded) reverse lexicographic order.
		There are many reverse lexicographic orders, in fact, there is one for each permutation of the variables.
		Note that this order is graded: it first orders by degree.
	www.math.sunysb.edu /~sorin/online-docs/Macaulay1-rel0994-html/node50.html (164 words)

	Info on Permutations
		In lexicographic order, the permutations are listed in numeric or dictionary order: 12...
		Those orders will have to be scheduled in order to meet deadlines, have material available, and satisfy other constraints.
		The order in which the orders are completed is a permutation.
	theory.cs.uvic.ca /~cos/amof/e_permI.htm (1346 words)

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