en>Erel Segal: /* Approximation and integrality gap */

2014-01-22T12:34:53Z

Approximation and integrality gap

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	~~Greetings~~. ~~Allow me start~~ by ~~telling you~~ the ~~author~~'s ~~title~~ - ~~Phebe~~. ~~Years ago~~ we ~~moved~~ to ~~North Dakota~~ and ~~I adore every day residing here~~. ~~What I adore doing~~ is ~~doing ceramics~~ but ~~I haven~~'~~t produced~~ a ~~dime~~ with it. ~~Hiring~~ is ~~my occupation~~.<br><br>~~My website~~ - ~~home std test kit~~ ([~~http:~~//~~www~~.~~videoworld~~.~~com~~/~~blog~~/~~211112 simply click~~ the ~~following website page~~])		In [[abstract algebra]], a '''heap''' (sometimes also called a '''groud'''<ref>Schein (1979) pp.101-102: footnote (o)</ref>) is a mathematical generalization of a [[group (mathematics)\|group]]. Informally speaking, a heap is obtained from a group by "forgetting" which element is the unit, in the same way that an [[affine space]] can be viewed as a [[vector space]] in which the 0 element has been "forgotten". A heap is essentially the same thing as a [[torsor]], and the category of heaps is equivalent to the category of torsors, with morphisms given by transport of structure under group homomorphisms, but the theory of heaps emphasizes the intrinsic composition law, rather than global structures such as the geometry of bundles.

			Formally, a heap is an [[structure (mathematical logic)\|algebraic structure]] consisting of a non-empty [[set (mathematics)\|set]] ''H'' with a [[ternary operation]] denoted <math>[x,y,z]\in H</math> which satisfies

			* the '''para-associative''' law

			::<math> [[a,b,c],d,e] = [a,[d,c,b],e] = [a,b,[c,d,e]] \ \forall \ a,b,c,d,e \in H</math>

			* the '''identity''' law

			::<math> [a,a,x] = [x,a,a] = x \ \forall \ a,x \in H. </math>

			A group can be regarded as a heap under the operation <math>[x,y,z] = xy^{-1}z </math>. Conversely, let ''H'' be a heap, and choose an element ''e'' ∈''H''. The [[binary operation]] <math>x*y = [x,e,y]</math> makes ''H'' into a group with [[identity element\|identity]] e and inverse <math> x^{-1} = [e,x,e] </math>. A heap can thus be regarded as a group in which the identity has yet to be decided.

			Whereas the [[automorphism]]s of a single object form a group, the set of [[isomorphism]]s between two isomorphic objects naturally forms a heap, with the operation <math>[f,g,h]=fg^{-1}h</math> (here juxtaposition denotes [[composition of functions]]). This heap becomes a group once a particular isomorphism by which the two objects are to be identified is chosen.

			==Examples==
			===Two element heap===
			If <math>H=\{a,b\}</math> then the following structure is a heap:
			:<math>[a,a,a]=a,\, [a,a,b]=b,\, [b,a,a]=b,\, [b,a,b]=a,</math>
			:<math>[a,b,a]=b,\, [a,b,b]=a,\, [b,b,a]=a,\, [b,b,b]=b.</math>

			===Heap of a group===
			As noted above, any group becomes a heap under the operation
			:<math>[x,y,z] = x y^{-1} z . </math>

			One important special case:
			====Heap of integers====
			If <math>x,y,z</math> are integers, we can set <math>[x,y,z]=x-y+z</math> to produce a heap. We can then choose any integer <math>k</math> to be the identity of a new group on the set of integers, with the operation <math>*</math>
			:<math>x*y = x+y-k</math>
			and inverse
			:<math>x^{-1} = 2k-x</math>.

			== Generalizations and related concepts ==

			* A '''pseudoheap''' or '''pseudogroud''' satisfies the partial para-associative condition<ref name=vagner1968>Vagner (1968)</ref>
			:<math>[[a,b,c],d,e] = [a,b,[c,d,e]] .</math>
			* A '''semiheap''' or '''semigroud''' is required to satisfy only the para-associative law but need not obey the identity law.<ref name=moldavska/>
			::An example of a semigroud that is not in general a groud is given by ''M'' a ring of matrices of fixed size with
			:::<math>[x,y,z] = x \cdot y^\top \cdot z</math>
			::where • denotes [[matrix multiplication]] and ⊤ denotes [[matrix transpose]].<ref name=moldavska>{{Cite journal \| last=Moldavs'ka \| first=Z. Ja. \| title=Linear semiheaps \| journal= Dopovidi Ahad. Nauk Ukrain. \| series=RSR Ser. A \| volume=1971 \| pages=888–890,957 \| id={{mathscinet\|45#6970}} \| postscript=<!--None--> }}</ref>
			* An '''idempotent semiheap''' is a semiheap where <math> [a,a,a] = a </math> for all ''a''.
			* A '''generalised heap''' or '''generalised groud''' is an idempotent semiheap where

			::<math> [a,a,[b,b,x]] = [b,b,[a,a,x]] </math> and <math> [[x,a,a],b,b] = [[x,b,b],a,a] </math> for all ''a'' and ''b''.

			A semigroud is a generalised groud if the relation → defined by

			:<math>a \rightarrow b \Leftrightarrow [a,b,a] = a </math>

			is [[reflexive relation\|reflexive]] (idempotence) and [[anti-symmetric relation\|anti-symmetric]]. In a generalised groud, → is an [[order relation]].<ref>Schein (1979) p.104</ref>

			* A [[torsor]] is an equivalent notion to a heap which places more emphasis on the associated group. Any <math>G</math>-torsor <math>X</math> is a heap under the operation <math>[x,y,z]=(x/y)\cdot z</math>. Conversely, if <math>X</math> is a heap, any <math>x,y\in X</math> define a [[permutation]] <math>\varphi_{x,y}(z)=[x,y,z]</math> of <math>X</math>. If we let <math>G</math> be the set of all such permutations <math>\varphi_{x,y}</math>, then <math>G</math> is a group and <math>X</math> is a <math>G</math>-torsor under the natural action.

			==Notes==
			{{reflist}}

			== References ==
			*{{cite book \| title=Twelve papers in logic and algebra \| first=Boris \| last=Schein \| authorlink = Boris Schein \| editor=A.F. Lavrik \| chapter=Inverse semigroups and generalised grouds \| series=Amer. Math. Soc. Transl. \| volume=113 \| publisher=[[American Mathematical Society]] \| year=1979 \| isbn=0-8218-3063-5 \| pages=89–182 }}
			*{{cite journal
			\| author = Vagner, V. V. \| authorlink=Viktor Wagner
			\| title = On the algebraic theory of coordinate atlases, II
			\| language = Russian
			\| journal = Trudy Sem. Vektor. Tenzor. Anal.
			\| volume = 14
			\| year = 1968
			\| pages = 229–281
			\| mr = 0253970}}

			[[Category:Algebraic structures]]

en>Praseodymium: Added link to NL wikipedia

2012-07-02T17:57:50Z

Added link to NL wikipedia

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Greetings. Allow me start by telling you the author's title - Phebe. Years ago we moved to North Dakota and I adore every day residing here. What I adore doing is doing ceramics but I haven't produced a dime with it. Hiring is my occupation.<br><br>My website - home std test kit ([http://www.videoworld.com/blog/211112 simply click the following website page])

Linear programming relaxation - Revision history

en>Erel Segal: /* Approximation and integrality gap */

en>Praseodymium: Added link to NL wikipedia