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Topic: Linearly ordered group

	PlanetMath: quasicyclic group
		In particular, this means that the subgroups are linearly ordered by inclusion, and all subgroups are fully invariant.
		The quasicyclic groups are the only infinite groups with a linearly ordered subgroup lattice.
		This is version 16 of quasicyclic group, born on 2005-11-25, modified 2006-03-23.
	planetmath.org /encyclopedia/QuasicyclicGroup.html (193 words)

	Ordered group - Wikipedia, the free encyclopedia
		In abstract algebra, an ordered group is a group G equipped with a partial order "≤" which is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then ag ≤ bg and ga ≤ gb.
		If G and H are two ordered groups, a map from G to H is a morphism of ordered groups if it is both a group homomorphism and a monotonic function.
		Ordered groups are used in the definition of valuations of fields.
	www.wikipedia.org /wiki/Ordered_group (336 words)

	Algebraic structure
		For example, the monster group simultaneously is an algebraic structure, and it has an algebraic structure: the structure shared by all groups.
		Linearly ordered group: a group with a compatible linear order
		Similarly, the category of topological groups (with continuous group homomorphisms as morphisms) is a category of topological spaces with extra structure.
	www.brainyencyclopedia.com /encyclopedia/a/al/algebraic_structure.html (1320 words)

	Jan Jakubik
		Jakubik, J., Lattice ordered groups with cyclic linearly ordered subgroups.
		Jakubik, J., On linearly ordered subgroups of a lattice ordered group.
		Jakubik, J.: On the congruence lattice of an abelian lattice ordered group.
	www.saske.sk /MI/eng/jakubik.htm (1776 words)

	Read about Ordered group at WorldVillage Encyclopedia. Research Ordered group and learn about Ordered group here! (Site not responding. Last check: 2007-10-22)
		partial order "≤" which is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤ b then ag ≤ bg and ga ≤ gb.
		subgroup of G is an ordered group: it inherits the order from G.
		Ordered groups are used in the definition of
	encyclopedia.worldvillage.com /s/b/Ordered_group (311 words)

	PlanetMath: total order
		A totally ordered set (or linearly ordered set) is a poset
		A totally ordered set is also sometimes called a chain, especially when it is considered as a subset of some other poset.
		This is version 16 of total order, born on 2001-10-06, modified 2006-11-25.
	planetmath.org /encyclopedia/Comparability.html (195 words)

	Stein Research Page 5
		Monodisperse spheres (polystyrene, poly(methylmethacrylate), silica) are close-packed iinto ordered arrays and infiltrated with a fluid which is solidified.
		After removal of the template, a solid skeleton is obtained around an ordered array of voids where the original spheres were located.
		The stop band position varies nearly linearly with refractive index of the penetrating fluid, and can be altered by adjusting the pore spacing, and the thickness, density, and composition of the walls—all parameters that we can control through chemistry.
	www.chem.umn.edu /groups/stein/researchtopic5.html (540 words)

	Chapter 2
		Second order cybernetics are sets of rule modification transforms relative to a specific class of situations, and relative to a set of consistent axioms with complete deductive span.
		A second order system is open in a way that is defined by what we have called, in Chapter 10, "semiotically open sets", ie, open via changes made through an interface between the first order system and a human decision maker or perhaps an complex computing device.
		The second order system must be assumed to be stable over the period under discussion, or else the temporal scale is not properly defined by the first order rates of change.
	www.bcngroup.org /area3/pprueitt/kmbook/Chapter2.htm (5463 words)

	Springer Online Reference Works
		A partially ordered set in which each two-element subset has both a least upper and a greatest lower bound.
		If a partially ordered set is regarded as a small category, then it is a lattice if and only if has products and coproducts of pairs of objects.
		Pudlák showed [a4] that this is closely related to a problem in finite group theory, which they solved for solvable groups.
	eom.springer.de /l/l057630.htm (996 words)

	Ordered group (Site not responding. Last check: 2007-10-22)
		In abstract algebra, an ordered group is a group G equipped with a partial order "≤" which is translation-invariant; in other words,"≤" has the property that, for all a, b, and g in G, if a ≤ bthen ag ≤ bg and ga ≤ gb.
		More generally, if G is an ordered group and X is some set, then the set of allfunctions from X to G is again an ordered group: all operations are performed componentwise.
		If G and H are two ordered groups, a map from G to H is a morphism of orderedgroups if it is both a group homomorphism and a monotonic function.
	www.therfcc.org /ordered-group-329375.html (221 words)

	Parsing using linearly ordered phonological rules
		Since the depth of ordering in a linguistically motivated description can easily approach 15-20, with many of the rules involved being many-ways ambiguous when regarded from the "wrong end," the approach of simply undoing the effects of the rules was soon seen to be quite impractical.
		However, with linearly ordered rules the number of adjacent consonants could in principle be increased by the application of certain rules preceding the deletion rule, including rules epenthesizing consonants, rules deleting vowels, and rules changing vowels into consonants.
		In most cases the linearly ordered rules were simpler than the two-level rules, in part because rule ordering rendered redundant some of the constraints necessitated by the two-level formalism.
	www.sil.org /computing/lascruces.html (5430 words)

	ABSTRACTS DROSTE
		The cofinality of a group G is the cardinality of the length of a shortest chain of proper subgroups terminating at G.
		Using combinatorial methods, we prove that in each of these lattices the partially ordered subset of all those elements which are finitely generated as normal subgroups is a lattice in which infima and suprema of subsets of cardinality $\leq\aleph$, always exist; two infinite distributive identities are also shown to hold.
		In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, we determine those groups which are equivalent in the sense that they may not be distinguished by the solvability of a system of finitely many equations in variables and parameters.
	www.math.tu-dresden.de /alg/droabal.html (3181 words)

	Muscle Wire Trade Study
		The group ordered a muscle wire sample kit containing 20 cm samples of wire with different diameters.
		The group is not experiencing any budget trouble, and although cost should be taken into account, it will not factor as heavily as other values.
		More wire than can linearly fit on the arm is needed to contract the necessary distance for finger actuation, so a method of wrapping or coiling the wire will need to fit on the forearm.
	www.nd.edu /~me463b25/Trade_Studies/TS_musclewire (2064 words)

	Any Group Will Do
		Let g be a linearly ordered abelian group, where the group operator respects order.
		Let the valuation of a member of r[g] be the lowest group element represented, according to the linear order of g.
		The "valuation" function is actually a group homomorphism from the nonzero elements of r[g] onto g.
	www.mathreference.com /id-val,anyg.html (649 words)

	Computational Biology Reading Group (Site not responding. Last check: 2007-10-22)
		We used the Clusters of Orthologous Groups database and information from published genomes to search for other universally conserved genes that have the same phylogenetic pattern as ribosomal RNA, and therefore constitute the ancestral genetic core of cells.
		Structured motifs may be described as an ordered collection of p > or = 1 "boxes" (each box corresponding to one part of the structured motif), p substitution rates (one for each box) and p - 1 intervals of distance (one for each pair of successive boxes in the collection).
		In particular, both algorithms time complexity scales linearly with N2n where n is the average length of the sequences and N their number.
	www.esat.kuleuven.ac.be /~saerts/cbrg.html (1272 words)

	Shaw & Gaines WCCCE 97 Paper
		In one group she is one of the supplier team, and in the other one of the customer team for a different project.
		Thus the two groups were able to communicate their designs and coordinate their negotiations using standard web tools in a simple and natural fashion.
		It is very inefficient for the entire group to participate in each aspect of the project, and meetings of the entire group are often long and tedious.
	www.cs.ubc.ca /wccce/program97/shaw/shaw.html (2378 words)

	Groups
		Each instance of #$Group is a composite object made up of one or more individual objects or events.
		#$Group in their own right that is some sort of specialization within GROUP, and (3) if a new member were to join SUBGROUP, by doing so it would also become a member of GROUP.
		A prototype exists prior to other members of the group; it may or may not itself be a member of the group (by default, Cyc assumes it is).
	www.cyc.com /cycdoc/vocab/group-vocab.html (2216 words)

	math lessons - Archimedean group (Site not responding. Last check: 2007-10-22)
		In abstract algebra, a branch of mathematics, an Archimedean group is an algebraic structure.
		We can also say that an Archimedean group is a linearly ordered group for which the Archimedean property holds.
		In the subsequent, we use the notation na (where n is in the set N of natural numbers) for the sum of a with itself n times.
	www.mathdaily.com /lessons/Archimedean_group (273 words)

	06: Order, lattices, ordered algebraic structures
		Linear orderings especially on infinite sets is the study of Ordinals in Set Theory; these are traditionally considered in 03: Mathematical Logic, especially 03G: Algebraic Logic.
		Ordered sets may be viewed as topological spaces; see 54: General Topology, especially 54F05: Ordered topological spaces, for more detail.
		There cannot be an ordering of the complex numbers consistent with the expected rules of arithmetic.
	www.math.niu.edu /~rusin/known-math/index/06-XX.html (606 words)

	Linearly ordered group - Wikipedia, the free encyclopedia
		In mathematics, a linearly ordered group is both a group and a linearly ordered set, in which the group operation is in a certain sense compatible with the linear ordering.
		For any x in the group G, either x ≥ 0 or −x ≥ 0, but not both, and
		Otto Hölder showed that every linearly ordered group satisfying an Archimedean property is isomorphic to a subgroup of the additive group of real numbers.
	en.wikipedia.org /wiki/Linearly_ordered_group (126 words)

	Encyclopedia: Archimedes
		After a number of experiments, whereby the hosts of the program tried burning a model wooden ship with a variety of mirrors, they concluded that the enemy ships would have had to have been virtually motionless and very close to shore for them to ignite, an unlikely scenario during a battle.
		A group at MIT subsequently performed their own tests and concluded that the mirror weapon was a possibility [2], although later tests of their system showed it to be ineffective in conditions that more closely matched the described siege [3].
		Archimedes was killed by a Roman soldier in the sack of Syracuse during the Second Punic War, despite orders from the Roman general, Marcellus, that he was not to be harmed.
	www.nationmaster.com /encyclopedia/Archimedes (5445 words)

	Going Backwards
		f is a field and g is an abelian group that is linearly ordered.
		If the valuation group g is isomorphic to the integers, then the image of an ideal h has a least element, and h is principle.
		This is called a discrete valuation group, and r is a discrete valuation ring, or dvr.
	www.mathreference.com /id-val,rev.html (1169 words)

	The Solar Wind Coupling and Geomagnetic Activity Group
		The group utilizes data acquired by past and present space missions as well as ground observatories to establish the morphology of these phenomena and to identify the physical processes that cause them.
		The group is also concerned with practical applications of these studies to the fields of scientific data management, geophysical exploration, and space weather forecasting.
		McPherron's group was one of the first to recognize that techniques such as linear prediction filtering could be used to encapsulate the behavior of the magnetosphere for prediction of activity measures, and that these encapsulations could be used to study the physical properties of the solar wind-magnetosphere system.
	www.igpp.ucla.edu /ssc/geomag/intro.html (1773 words)

	[No title]
		Then $X$ is called a {\bf linearly ordered set} if: \begin{enumerate} \item[(i)] $x\le x$ for all $x\in X$ \item[(ii)] if $x\le y$ and $y\le x$ then $x=y$ \item[(iii)] if $x\le y$ and $y\le z$ then $x\le z$ \item[(iv)] for any two $x,y\in X$, either $x\le y$ or $y\le x$.
		The phrase totally ordered set is sometimes used to mean the same thing as "linearly ordered set".
		A set with a relation which satisfies the first three rules for a linearly ordered set is called a partially ordered set.
	www.math.unl.edu /~webnotes/src/classes-1995/class02.wfy (868 words)

	Property Adition Commutative >> Associative Additon Property Identity - Equality Adition Property (Site not responding. Last check: 2007-10-22)
		In mathematics (particularly abstract algebra), the Archimedean property is a property held by some ordered algebraic structures, and in particular by the ordered field of real numbers.
		In particular, a linearly ordered group that is Archimedean is an Archimedean group, and an ordered field that is Archimedean is an Archimedean field.
		If x and y are positive numbers (or positive, non-zero elements of any ordered algebraic structure), then x is infinitesimal with respect to y (or equivalently, y is infinite with respect to x) if, for every natural number n, the multiple nx is less than y.
	www.7newsty.info /education/addition-property.htm (307 words)

	Concurrency Abstracts (Site not responding. Last check: 2007-10-22)
		In models of concurrent processes constraints on the order of events are often represented by partial orders, and schedules of events are then defined using an algebra of standard operations such as sequential and parallel composition.
		Partial orders are the simple case of a monoid whose two elements represent the presence or absence of a constraint.
		The heart of the approach is a notion of partial string derived from the view of a string as a linearly ordered multiset by relaxing the linearity constraint, thereby permitting partially ordered multisets or pomsets.
	boole.stanford.edu /abstracts.html (9620 words)

	Algebraic structure - ExampleProblems.com
		For example, a group (G,) is usually referred simply as a group* G.
		For example, a topological group is a topological space with a group structure such that the operations of multiplication and taking inverses are continuous; a topological group has both a topological and an algebraic structure.
		Every algebraic structure has its own notion of homomorphism, a function that is compatible with the given operation(s).
	www.exampleproblems.com /wiki/index.php?title=Algebraic_structure&printable=yes (607 words)

	William Howe on Bruce Andrews
		The words are grouped around the axis like the two dimensional projection of a three dimensional model of a helical molecule.
		We are set up for a certain group of sounds by "lathering," and as we read those sounds recur in the following words in a very controlled way, and as a consequence the first five words function as a kind of acoustic sentence.
		What you are reading now is a linearly ordered group of words that "add up" to a larger discursive group that "means" independently of (or maybe codependently with) the individual words themselves.
	epc.buffalo.edu /authors/andrews/about/howe.html (8587 words)

	Keimer's Group (Site not responding. Last check: 2007-10-22)
		Here the space near the reactor wall is very limited and the monochromator shielding has to be placed at the end of a neutron guide at a distance of about 10 m from the wall in order to gain enough space for the spectrometer.
		The optimal settings calculated from the known phonon data (energy and group velocity) are indicated by arrows.
		The resonance in the line width is expected in a region where the group velocity is higher than the velocity of sound (Pitaevskii broadening).
	www.fkf.mpg.de /keimer/Nspectro/nspectro.html (2947 words)

	Permutation Group Problems (Site not responding. Last check: 2007-10-22)
		The Parker vector of a finite permutation group G is the n-tuple whose kth component is the number of orbits of G on the set of k-cycles occurring in elements of G.
		Pablo Spiga found that the group of order 128 with generators (1,2,7,3)(4,8)(5,11)(6,9)(10,14)(12,16,13,15), (1,4,7,5)(2,8)(3,11)(6,13,14,12)(9,15)(10,16), and (1,6)(2,9,3,10)(4,12,5,13)(7,14)(8,15)(11,16) is a counterexample.
		Eamonn O'Brien has shown that there are two pairs of groups of order 64 (numbers 19 and 111, and 94 and 249, in the lists in MAGMA and GAP), which act transivitely on 16 points, such that the two groups in each pair give rise to the same sets of cycle partitions.
	www.maths.qmw.ac.uk /~pjc/pgprob.html (2622 words)

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