Great circle - Revision history

en>Fgnievinski: /* See also */

2014-08-18T03:07:26Z

See also

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I wonder which~~ of ~~the companies will be around the next 2 decades at AT & T's strangle hold our national market~~. For your kitchen at home, it's great if you will find multiple families sharing the identical space, or for the people with home businesses. The equipment and DVR they feature is part with the monthly subscription and valuable to the customer who would like to record a show and observe it later on.<br><br>Is a must manage the [http://~~cloud~~.~~globe~~.~~gov~~/~~de/web/jmackaro/blog/-/blogs/6673693;jsessionid=3A07519D746894DAF0081F791D69895A~~?~~_33_redirect=~~http://cctvdvrreviews.com open source dvr software for cctv] workers should they perform their job properly. These cameras are available in a myriad of designs, technologies and colours. Recording over a scheduled basis allows to save disk space by recording only if needed. 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en>Incnis Mrsi: /* top */ the syntax apparently prompted the red-faced people to alter wording back and forth for no good

2014-02-18T16:54:34Z

top: the syntax apparently prompted the red-faced people to alter wording back and forth for no good

Show changes

en>Fgnievinski: /* Applications */

2013-12-25T14:46:42Z

Applications

← Older revision		Revision as of 16:46, 25 December 2013
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	~~Computer Community~~ and ~~Systems Engineer Hackley from Lorraine~~, ~~loves~~ to ~~spend time mountain climbing~~, ~~swiss watches replica~~ and ~~antiques~~. ~~Finds~~ the ~~world~~ an ~~fascinating place having spent 4 months at Ironbridge Gorge~~.<br><br>~~Also visit my blog~~; ~~tuttogratissss~~.~~altervista~~.~~org~~ ([~~http~~://~~tuttogratissss~~.~~altervista~~.~~org~~/~~datingclub~~/?L=~~blogs~~.~~blog~~&~~article~~=~~16643 please click the next website~~])		{{For\|the term as used in elementary geometry\|congruence (geometry)}}

			In [[abstract algebra]], a '''congruence relation''' (or simply '''congruence''') is an [[equivalence relation]] on an [[algebraic structure]] (such as a [[group (mathematics)\|group]], [[ring (mathematics)\|ring]], or [[vector space]]) that is compatible with the structure. Every congruence relation has a corresponding [[quotient]] structure, whose elements are the [[equivalence class]]es (or '''congruence classes''') for the relation.

			==Basic example==

			The prototypical example of a congruence relation is [[Modular arithmetic#Congruence relation\|congruence modulo <math>n</math>]] on the set of [[integer]]s. For a given [[positive integer]] <math>n</math>, two integers <math>a</math> and <math>b</math> are called '''congruent modulo <math>n</math>''', written
			: <math>a \equiv b \pmod{n}</math>
			if <math>a - b</math> is [[divisible]] by <math>n</math> (or equivalently if <math>a</math> and <math>b</math> have the same [[remainder]] when divided by <math>n</math>).

			for example, <math>37</math> and <math>57</math> are congruent modulo <math>10</math>,

			: <math>37 \equiv 57 \pmod{10}</math>

			since <math>37 - 57 = -20</math> is a multiple of 10, or equivalently since both <math>37</math> and <math>57</math> have a remainder of <math>7</math> when divided by <math>10</math>.

			Congruence modulo <math>n</math> (for a fixed <math>n</math>) is compatible with both [[addition]] and [[multiplication]] on the integers. That is, if

			: <math>a_1 \equiv a_2 \pmod{n}</math> and <math>b_1 \equiv b_2 \pmod{n}</math>

			then

			: <math>a_1 + b_1 \equiv a_2 + b_2 \pmod{n}</math> and <math>a_1 b_1 \equiv a_2b_2 \pmod{n}</math>

			The corresponding addition and multiplication of equivalence classes is known as [[modular arithmetic]]. From the point of view of abstract algebra, congruence modulo <math>n</math> is a congruence relation on the [[ring (mathematics)\|ring]] of integers, and arithmetic modulo <math>n</math> occurs on the corresponding [[quotient ring]].

			==Definition==
			The definition of a congruence depends on the type of [[algebraic structure]] under consideration. Particular definitions of congruence can be made for [[group (mathematics)\|groups]], [[ring (mathematics)\|rings]], [[vector space]]s, [[module (mathematics)\|modules]], [[semigroup]]s, [[lattice (order)\|lattices]], and so forth. The common theme is that a congruence is an [[equivalence relation]] on an algebraic object that is compatible with the algebraic structure, in the sense that the [[Operation (mathematics)\|operations]] are [[well-defined]] on the [[equivalence classes]].

			For example, a group is an algebraic object consisting of a [[set (mathematics)\|set]] together with a single [[binary operation]], satisfying certain axioms. If <math>G</math> is a group with operation &lowast;, a '''congruence relation''' on ''G'' is an equivalence relation &equiv; on the elements of ''G'' satisfying
			:''g''<sub>1</sub> &equiv; ''g''<sub>2</sub>  and  ''h''<sub>1</sub> &equiv; ''h''<sub>2</sub>    ⇒    ''g''<sub>1</sub> &lowast; ''h''<sub>1</sub> &equiv; ''g''<sub>2</sub> &lowast; ''h''<sub>2</sub>
			for all ''g''<sub>1</sub>, ''g''<sub>2</sub>, ''h''<sub>1</sub>, ''h''<sub>2</sub> ∈ ''G''. For a congruence on a group, the equivalence class containing the [[identity element]] is always a [[normal subgroup]], and the other equivalence classes are the [[coset]]s of this subgroup. Together, these equivalence classes are the elements of a [[quotient group]].

			When an algebraic structure includes more than one operation, congruence relations are required to be compatible with each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on a ring must satisfy
			:''r''<sub>1</sub> + ''s''<sub>1</sub> &equiv; ''r''<sub>2</sub> + ''s''<sub>2</sub>    and    ''r''<sub>1</sub>''s''<sub>1</sub> &equiv; ''r''<sub>2</sub>''s''<sub>2</sub>
			whenever ''r''<sub>1</sub> &equiv; ''r''<sub>2</sub> and ''s''<sub>1</sub> &equiv; ''s''<sub>2</sub>. For a congruence on a ring, the equivalence class containing 0 is always a two-sided [[Ideal (ring theory)\|ideal]], and the two operations on the set of equivalence classes define the corresponding [[quotient ring]].

			The general notion of a congruence relation can be given a formal definition in the context of [[universal algebra]], a field which studies ideas common to all [[algebraic structures]]. In this setting, a congruence relation is an equivalence relation &equiv; on an algebraic structure that satisfies
			:''μ''(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub>) &equiv; ''μ''(''a''<sub>1</sub>′, ''a''<sub>2</sub>′, ..., ''a''<sub>''n''</sub>′)
			for every ''n''-ary operation ''μ'', and all elements ''a''<sub>1</sub>,...,''a''<sub>''n''</sub>,''a''<sub>1</sub>′,...,''a''<sub>''n''</sub>′ satisfying ''a''<sub>''i''</sub> &equiv; ''a''<sub>''i''</sub>′ for each ''i''.

			==Relation with homomorphisms==
			If &fnof;: ''A'' → ''B'' is a [[homomorphism]] between two algebraic structures (such as [[group homomorphism\|homomorphism of groups]], or a [[linear map]] between [[vector space]]s), then the relation &equiv; defined by
			:''a''<sub>1</sub> &equiv; ''a''<sub>2</sub>    if and only if    &fnof;(''a''<sub>1</sub>) = &fnof;(''a''<sub>2</sub>)
			is a congruence relation. By the [[first isomorphism theorem]], the [[image (mathematics)\|image]] of ''A'' under &fnof; is a substructure of ''B'' [[Isomorphism\|isomorphic]] to the quotient of ''A'' by this congruence.

			==Congruences of groups, and normal subgroups and ideals==
			In the particular case of [[group (mathematics)\|groups]], congruence relations can be described in elementary terms as follows:
			If ''G'' is a group (with [[identity element]] ''e'' and operation *) and ~ is a [[binary relation]] on ''G'', then ~ is a congruence whenever:
			#[[Given any]] element ''a'' of ''G'', ''a'' ~ ''a'' ('''[[Reflexive relation\|reflexivity]]''');
			#Given any elements ''a'' and ''b'' of ''G'', [[material conditional\|if]] ''a'' ~ ''b'', then ''b'' ~ ''a'' ('''[[Symmetric relation\|symmetry]]''');
			#Given any elements ''a'', ''b'', and ''c'' of ''G'', if ''a'' ~ ''b'' [[logical conjunction\|and]] ''b'' ~ ''c'', then ''a'' ~ ''c'' ('''[[Transitive relation\|transitivity]]''');
			#Given any elements ''a'', ''a' '', ''b'', and ''b' '' of ''G'', if ''a'' ~ ''a' '' and ''b'' ~ ''b' '', then ''a'' * ''b'' ~ ''a' '' * ''b' '';
			#Given any elements ''a'' and ''a' '' of ''G'', if ''a'' ~ ''a' '', then ''a''<sup>−1</sup> ~ ''a' ''<sup>−1</sup> (this can actually be proven from the other four, so is strictly redundant).

			Conditions 1, 2, and 3 say that ~ is an [[equivalence relation]].

			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]].
			Specifically, ''a'' ~ ''b'' if and only if ''b''<sup>−1</sup> * ''a'' ~ ''e''.
			So instead of talking about congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruence corresponds uniquely to some normal subgroup of ''G''.

			=== Ideals of rings and the general case ===

			A similar trick allows one to speak of kernels in [[ring (mathematics)\|ring theory]] as [[ideal (ring theory)\|ideals]] instead of congruence relations, and in [[module (mathematics)\|module theory]] as [[submodule]]s instead of congruence relations.

			The most general situation where this trick is possible is with [[Omega-group]]s (in the general sense allowing operators with multiple arity). But this cannot be done with, for example, [[monoid]]s, so the study of congruence relations plays a more central role in monoid theory.

			==Universal algebra==

			The idea is generalized in [[universal algebra]]:
			A congruence relation on an algebra ''A'' is a [[subset]] of the [[direct product]] ''A'' × ''A'' that is both an [[equivalence relation]] on ''A'' and a [[subalgebra]] of ''A'' × ''A''.

			The [[kernel (universal algebra)\|kernel]] of a [[homomorphism]] is always a congruence. Indeed, every congruence arises as a kernel.
			For a given congruence ~ on ''A'', the set ''A''/~ of [[equivalence class]]es can be given the structure of an algebra in a natural fashion, the [[quotient algebra]].
			The function that maps every element of ''A'' to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.

			The [[lattice (order)\|lattice]] '''Con'''(''A'') of all congruence relations on an algebra ''A'' is [[algebraic lattice\|algebraic]].

			==See also==
			*[[Table of congruences]]
			*[[Linear congruence theorem]]
			*[[Congruence lattice problem]]

			==References==
			* Horn and Johnson, ''Matrix Analysis,'' Cambridge University Press, 1985. ISBN 0-521-38632-2. (Section 4.5 discusses congruency of matrices.)

			{{DEFAULTSORT:Congruence Relation}}
			[[Category:Modular arithmetic]]
			[[Category:Algebra]]
			[[Category:Mathematical relations]]

en>Mindmatrix: tweak

2012-08-19T14:35:58Z

tweak

New page

Computer Community and Systems Engineer Hackley from Lorraine, loves to spend time mountain climbing, swiss watches replica and antiques. Finds the world an fascinating place having spent 4 months at Ironbridge Gorge.<br><br>Also visit my blog; tuttogratissss.altervista.org ([http://tuttogratissss.altervista.org/datingclub/?L=blogs.blog&article=16643 please click the next website])