en>Jordgette: /* Coordinate transformation */ Let's not mix events and people

2013-12-08T22:20:39Z

Coordinate transformation: Let's not mix events and people

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	~~Friends call him Royal Seyler~~. ~~She works~~ as a ~~financial officer~~ and ~~she will~~ not ~~change it anytime soon~~. ~~Playing crochet~~ is some ~~thing~~ that I'~~ve done for many years~~. ~~I currently live~~ in ~~Alabama~~.<br><br>~~my blog post~~ [~~http~~://~~www~~.~~kindkids~~.~~net~~/~~ActivityFeed~~/~~MyProfile~~/~~tabid~~/61/~~UserId~~/~~260741~~/~~Default~~.~~aspx extended auto warranty~~]		{{Confusing\|date=March 2011}}
			{{Expert-subject\|Mathematics\|date=February 2009}}

			In [[algebra]], a '''presentation of a monoid''' (or '''semigroup''') is a description of a [[monoid]] (or [[semigroup]]) in terms of a set Σ of generators and a set of relations on the [[free monoid]] Σ<sup>∗</sup> (or free semigroup Σ<sup>+</sup>) generated by Σ. The monoid is then presented as the [[quotient monoid\|quotient]] of the free monoid by these relations. This is an analogue of a [[group presentation]] in [[group theory]].

			As a mathematical structure, a monoid presentation is identical to a [[string rewriting system]] (also known as semi-Thue system). Every monoid may be presented by a semi-Thue system (possibly over an infinite alphabet).<ref>Book and Otto, Theorem 7.1.7, p. 149</ref>

			A ''presentation'' should not be confused with a ''[[Representation theory\|representation]]''.

			== Construction ==

			The relations are given as a (finite) [[binary relation]] ''R'' on Σ<sup>∗</sup>. To form the quotient monoid, these relations are extended to [[monoid congruence]]s as follows.

			First, one takes the symmetric closure ''R'' ∪ ''R''<sup>−1</sup> of ''R''. This is then extended to a symmetric relation ''E'' ⊂ Σ<sup>∗</sup> × Σ<sup>∗</sup> by defining ''x'' ~<sub>''E''</sub> ''y'' if and only if ''x'' = ''sut'' and ''y'' = ''svt'' for some strings ''u'', ''v'', ''s'', ''t'' ∈ Σ<sup>∗</sup> with (''u'',''v'') ∈ ''R'' ∪ ''R''<sup>−1</sup>. Finally, one takes the reflexive and transitive closure of ''E'', which is then a monoid congruence.

			In the typical situation, the relation ''R'' is simply given as a set of equations, so that <math>R=\{u_1=v_1,\cdots,u_n=v_n\}</math>. Thus, for example,
			:<math>\langle p,q\,\vert\; pq=1\rangle</math>
			is the equational presentation for the [[bicyclic monoid]], and

			:<math>\langle a,b \,\vert\; aba=baa, bba=bab\rangle</math>
			is the [[plactic monoid]] of degree 2 (it has infinite order). Elements of this plactic monoid may be written as <math>a^ib^j(ba)^k</math> for integers ''i'', ''j'', ''k'', as the relations show that ''ba'' commutes with both ''a'' and ''b''.

			==Inverse monoids and semigroups==


			<!-- this stuff is correct, it just misses the definitions of some basic concepts found in Howie, like the minimum congruence (on a semigroup) containing a given relation and so on. Wagner congruence is no longer a red link now. The presentation could be made easier to digest and less ladden with notations -->

			Presentations of inverse monoids and semigroups can be defined in a similar way using a pair
			:<math>(X;T)</math>
			where

			<math> (X\cup X^{-1})^* </math>

			is the [[free monoid with involution]] on <math>X</math>, and

			:<math>T\subseteq (X\cup X^{-1})^\times (X\cup X^{-1})^</math>

			is a [[binary function\|binary]] relation between words. We denote by <math>T^{\mathrm{e}}</math> (respectively <math>T^\mathrm{c}</math>) the [[equivalence relation]] (respectively, the [[congruence relation\|congruence]]) generated by ''T''.

			We use this pair of objects to define an inverse monoid

			:<math>\mathrm{Inv}^1 \langle X \| T\rangle.</math>

			Let <math>\rho_X</math> be the [[Wagner congruence]] on <math>X</math>, we define the inverse monoid

			:<math>\mathrm{Inv}^1 \langle X \| T\rangle</math>

			''presented'' by <math>(X;T)</math> as

			:<math>\mathrm{Inv}^1 \langle X \| T\rangle=(X\cup X^{-1})^*/(T\cup\rho_X)^{\mathrm{c}}.</math>

			In the previous discussion, if we replace everywhere <math>({X\cup X^{-1}})^*</math> with <math>({X\cup X^{-1}})^+</math> we obtain a '''presentation (for an inverse semigroup)''' <math>(X;T)</math> and an inverse semigroup <math>\mathrm{Inv}\langle X \| T\rangle</math> '''presented''' by <math>(X;T)</math>.

			A trivial but important example is the '''free inverse monoid''' (or '''free inverse semigroup''') on <math>X</math>, that is usually denoted by <math>\mathrm{FIM}(X)</math> (respectively <math>\mathrm{FIS}(X)</math>) and is defined by

			:<math>\mathrm{FIM}(X)=\mathrm{Inv}^1 \langle X \| \varnothing\rangle=({X\cup X^{-1}})^*/\rho_X,</math>
			or
			:<math>\mathrm{FIS}(X)=\mathrm{Inv} \langle X \| \varnothing\rangle=({X\cup X^{-1}})^+/\rho_X.</math>

			==Notes==
			{{Reflist}}

			==References==
			* John M. Howie, ''Fundamentals of Semigroup Theory'' (1995), Clarendon Press, Oxford ISBN 0-19-851194-9
			* M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7.
			* [[Ronald V. Book]] and Friedrich Otto, ''String-rewriting Systems'', Springer, 1993, ISBN 0-387-97965-4, chapter 7, "Algebraic Properties"

			{{DEFAULTSORT:Presentation Of A Monoid}}
			[[Category:Semigroup theory]]

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

2012-07-19T00:29:14Z

Robot - Moving category Fundamental physics concepts to Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

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Friends call him Royal Seyler. She works as a financial officer and she will not change it anytime soon. Playing crochet is some thing that I've done for many years. I currently live in Alabama.<br><br>my blog post [http://www.kindkids.net/ActivityFeed/MyProfile/tabid/61/UserId/260741/Default.aspx extended auto warranty]

Introduction to mathematics of general relativity - Revision history

en>Jordgette: /* Coordinate transformation */ Let's not mix events and people

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.