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Topic: Congruence relation

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	Congruence relation - Wikipedia, the free encyclopedia
		In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s).
		Congruences typically arise as kernels of homomorphisms, and in fact every congruence is the kernel of some homomorphism: For a given congruence ~ on A, the set A/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra.
		Notice that such a congruence ~ is determined entirely by the set {a ∈ G : a ~ e} of those elements of G that are congruent to the identity element, and this set is a normal subgroup.
	en.wikipedia.org /wiki/Congruence_relation (645 words)

	Congruence - Wikipedia, the free encyclopedia
		In the theory of smooth manifolds, especially in the context of general relativity, congruence refers to the integral curves defined by a vector field.
		In psychology and NLP, congruence could be defined as rapport within oneself, or internal and external consistency, perceived as sincerity.
		In cladistics, congruence is a test of homology, or shared, derived character states, in which the distributions of supposed homologies among taxa are compared for consistency.
	en.wikipedia.org /wiki/Congruence (164 words)

	Congruence (geometry) - Wikipedia, the free encyclopedia
		In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis.
		Two triangles are congruent if their corresponding sides and angles are equal.
		If the hypotenuse and a certain leg of a triangle are congruent to the corresponding hypotenuse and leg of a different triangle, the two triangles are congruent.
	en.wikipedia.org /wiki/Congruence_(geometry) (426 words)

	Alfred North Whitehead: The Concept of Nature: Chapter 6: Congruence (Site not responding. Last check: 2007-11-07)
		Thus immediate judgments of congruence are presupposed in measurement, and the process of measurement is merely a procedure to extend the recognition of congruence to cases where these immediate judgments are not available.
		It has then been proved that there are alternative relations which satisfy these conditions equally well and that there is nothing intrinsic in the theory of space to lead us to adopt any one of these relations in preference to any other as the relation of congruence which we adopt.
		Congruence depends on motion, and thereby is generated the connexion between spatial congruence and temporal congruence.
	spartan.ac.brocku.ca /~lward/Whitehead/Whitehead_1920/White1_06.html (6039 words)

	Congruence and Similarity
		Congruent triangles are a special type of similar triangles.
		Congruent triangles have the same shape (similar triangles) and size.
		DE The factor for congruent triangles is 1.
	argyll.epsb.ca /jreed/math9/strand3/3202.htm (578 words)

	Lines and Boxes
		The relation defined by the reaction rules (which we will call the 'reaction relation') defines only how two or more processes interact with one another; they do not explain how actions can be exposed to the environment.
		The relation defined by the transition rules (the 'transition relation') defines the full labeled transition system for the set of processes (similar to a Kripke structure for the system).
		This relation is an extension of the reaction relation; that is, it contains all transitions in the reaction relation plus some additional ones.
	www.ccs.neu.edu /home/matthias/369-s04/Transcripts/pi-2-summary.html (986 words)

	Robert Rosen - The Modeling Relation
		As Rosen says: "...indeed the modelling relation is a ubiquitous characteristic of everyday life as well as science." [5a] Therefore, when we subjectively define a natural system, we are engaging in a mental modeling act.
		Further, it should be noted that even percepts are not to be regarded as some kind of direct knowing of the external world; but rather, percepts are themselves a kind of mental model ("sensory impressions") of qualities we can only deem as plausible to actually have their source in the external world.
		So, the congruence relation established is only between the elements and relations specified in the formal model and a corresponding certain finite number of observables and relations in the material realization.
	www.panmere.com /rosen/faq_mr1.htm (2393 words)

	Equivalence Relations
		The basic idea of a strong bisimulation relation is that two systems should be able to pass through equivalent states, such that one can simulate the others' behaviour at all times.
		This is the useful mid-point between strong and weak bisimulation relations that is a congruence.
		Forming these relations is an iterative process, starting with the two systems initial states, and for each transition that each system can make, relating the resulting state to another state in the other system.
	www.cs.bris.ac.uk /~edwards/prep2001/html/node9.html (534 words)

	L10.html
		Recall the definition of congruence: We say x is congruent to y modulo m and we write this as x=y (mod m).
		Also recall that the congruence relation is an equivalence relation.
		Definition: A congruence class modulo m is the set of all integers congruent to a fixed integer k, where m is a natural number.
	www.math.sfu.ca /~gfee/Math342/L101.html (833 words)

	Congruence (geometry) (Site not responding. Last check: 2007-11-07)
		In geometry, two shapes are called congruent if one can be transformed into the other by a series of translationss, rotations and reflections.
		The third is a different size, and so is similar but not congruent to the first two; the fourth is different altogether.
		Two triangles are congruent if their corresponding sides and angles are equal in measure.
	www.sciencedaily.com /encyclopedia/congruence__geometry_ (386 words)

	CS342: Automata (Site not responding. Last check: 2007-11-07)
		L is the union of some congruence classes of congruence relation R implies that L is the union of some of the equivalence classes of an equivalence relation of finite index.
		So, if L is the union of some congruence classes of congruence relation of finite index, then L is regular.
		Since congruence relation R refines RM, then L is the union of some congruence classes of congruence relation R. So, if L is regular then it is the union of some congruence classes of congruence relation of finite index.
	www.cs.nmt.edu /~cs342/Exam2.htm (914 words)

	Operational and Algebraic Semantics of Concurrent Processes
		Rather we adopt the approach by which the lambda calculus was first introduced; we begin by defining a small language whose constructions reflect simple operational ideas, and then give the language meaning as a transition system by means of inference rules for evaluation.
		Thereafter we define a congruence relation over the language, called observation-congruence, based upon the idea of bisimulation from David Park.
		This congruence relation is therefore interpreted as equality of processes.
	www.lfcs.inf.ed.ac.uk /reports/88/ECS-LFCS-88-46 (199 words)

	[No title]
		This is then a congruence relation and the equivalence classes might form an algebra which is then a free algebra.
		Complications can occur when the congruence classes are not sufficient, for instance because the axioms require the existence of certain things which are not fulfilled by any of these terms.
		In this case, the congruences are determined by equations that hold for all members of the class.
	www.math.niu.edu /~rusin/known-math/00_incoming/free_alg (3067 words)

	v_congr
		In a group, a congruence relation is the same thing as the coset decomposition for a normal subgroup.
		In a commutative ring, a congruence relation is the same thing as the coset decomposition for an ideal.
		This is a contrast with the specific cases of groups and rings, where the kernel is a normal subgroup.
	www.math.ucla.edu /~baker/222a/handouts/v_congr/node3.html (106 words)

	Articles - Modular arithmetic (Site not responding. Last check: 2007-11-07)
		Two integers a, b are said to be congruent modulo n if their difference is divisible by n.
		This is the prototypical example of a congruence relation.
		The notion of modular arithmetic is related to that of the remainder in division.
	lastring.com /articles/Modular_arithmetic?mySession=7d59852336dc02f3... (852 words)

	Course Notes for COMP4151: Comparative Concurrency Semantics
		I explained what it means for an equivalence relation to be a congruence for an operator, or, in other words, for the operator to be compositional for the equivalence.
		However, completed trace equivalence is not a congruence for the synchronous parallel composition of CSP and for the restriction operators of CCS and ACP.
		Theorem: Given an equivalence relation ~ on a domain D (a set of mathematical objects such as process graphs), and given a number of operators on D, there exists a coarsest congruence relation that is finer than ~.
	www.cse.unsw.edu.au /~rvg/AdvancedTopicsInConcurrency/notes.html (2788 words)

	h13
		The relation is reflexive but not symmetric, asymmetric, antisymmetric, or irreflexive.
		The relation is asymmetric and antisymmetric, but not reflexive or irreflexive.
		Write the the relation using ordered pair notation, and show that the relation is antisymmetric.
	www.cs.umd.edu /class/fall2001/cmsc250/HW/h13/h13.html (223 words)

	Footnotes (Site not responding. Last check: 2007-11-07)
		When f is a homomorphism this equivalence relation is called a congruence relation.
		A related story: Bjarni Jónsson used lattice theory to define quasi-isomorphism of Abelian groups and to show that, under this notion, the Krull-Schmidt Theorem on the uniqueness of direct decompositions, held for torsion-free Abelian groups of finite rank.
		A month later I was talking with Lee Lady who told me that Jónsson did the Abelian group community a great favor by proving his theorem with lattice theory: it forced them to reformulate and reprove it and thereby understand it much better.
	www.math.hawaii.edu /~ralph/schmidt/sch-protter/footnode.html (172 words)

	Department of Computer Science - The University of Iowa
		We view the congruence closure problem as a combination problem and apply the variable abstraction technique to it to obtain the notion of an abstract congruence closure.
		If we consider the rewrite relation, rather than the congruence relation, we get the notion of a tree automata and an abstract rewrite closure.
		The combination of congruence closure and rewrite closure can be used to solve a long-standing open problem in term rewriting.
	www.cs.uiowa.edu /Events/Colloquia/2002/03-01.html (332 words)

	Articles - Equivalence class (Site not responding. Last check: 2007-11-07)
		In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:
		This equivalence relation is known as the kernel of f.
		Because of the properties of an equivalence relation it holds that a is in [a] and that any two equivalence classes are either equal or disjoint.
	www.gaple.com /articles/Equivalence_class?mySession=3ee23dcf728a1d0197e9ad78219f2610 (842 words)

	dd_jonsson
		An ``ultraproduct'' of algebras is their direct product modulo a congruence relation constructed from a nonprincipal ultrafilter.
		The congruence relation tends to collapse the product down to something that looks like a ``generic'' copy of the individual algebras, reflecting whatever features they have in common.
		The construction is set-theoretic and actually works for sets with relations as well as for algebras.
	www.math.ucla.edu /~baker/222a/handouts/dd_jonsson/node4.html (175 words)

	v_congr
		This congruence is an equivalence relation that is compatible with the ring operations, in the following sense:
		Also, we may say ``congruence'' instead of ``congruence relation''.
		Just as for equivalence relations in general, we can speak of the blocks of a congruence relation (or ``classes'', but that usage is somewhat old).
	www.math.ucla.edu /~baker/222a/handouts/v_congr/node2.html (82 words)

	Relations
		The relation of divisibility is a partial order on the set S = Div (n).
		, then the relation "a is congruent to b (mod n)" is an equivalence relation.
		The equivalence classes of the relation "congruence module n" are called the
	www.lv.psu.edu /OJJ/courses/ist-230/topics/relations.html (659 words)

	class21
		so that every pair of elements from the class is in the relation, while any pair of an element in the class and an element outside the class is not in the relation.
		, the following is an equivalence relation on
		etc. are the equivalence classes of the relation ``congruence modulo
	www.mscs.dal.ca /~janssen/2113/notes/class21/class21.html (67 words)

	Class Notes 10 for Mth341 (Site not responding. Last check: 2007-11-07)
		In high school geometry congruence is often defined as “equal in measure”.
		That means two angles are defined to be congruent if they have the same measures, two segments are defined to be congruent if they have the same lengths, two triangles are defined to be congruent if the correspondent angles and segments are equal in measure, etc.
		Given two points A and B. A point C is equidistant from A and B if and only iff C is on the perpendicular bisector of the segment AB.
	www.emunix.emich.edu /~gisela/M341/classnotes/classnotes10.html (188 words)

	2. Basic Definitions
		is said to be a congruence relation in case it is admissible, i.e., compatible with concatenation.
		The following definitions are closely related to describing monoids and groups in terms of generators and defining relations.
		We will later on see how such representations are related to different reductions, which will be Noetherian because of the following statements, which heavily depend on the presentation of the group.
	www.mathematik.uni-kl.de /~zca/Reports_on_ca/09/paper_html/node2.html (1523 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		Topics include the congruence relation, arithmetic functions, Gauss’ Law of Quadratic Reciprocity, and Diophantine equations as well as applications such as cryptography.
		Recognize the historical importance of the congruence relation in number theory.
		Congruences (9 hours) Congruences, linear congruences, the Chinese Remainder Theorem, Wilson’s Theorem, Fermat’s Little Theorem, and Euler’s Theorem 3.
	www.lhup.edu /ucc/mathematics/MATH302_rev.doc (648 words)

	Finitism in Geometry
		If space and time are discrete, then the runner, the tortoise, Achilles and all other moving objects simply go through a finite number of space locations in a finite number of time elements and all the problems with supertasks vanish as a one-minute time interval is no longer divisible in a denumerable series of intervals.
		In the very same sense, there is a clear difference with all the theories and proposals that have been put forward in the computer sciences.
		In recent years some authors have focused rather on particular problems related to strict finitist geometry instead of attempting to present a full alternative.
	plato.stanford.edu /entries/geometry-finitism (4873 words)

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