Theory of oriented matroids and convexity J.L. Ram´ ırez Alfons´ ın IMAG, Universit´ e de Montpellier CombinatoireS , Summer School , Paris, June 29 - July 3 2015 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
A signed set X is a set X partitionned in two parts ( X + , X − ), where X + is the set of positive elements of X and X − is the set of negatives elements. The set X = X + ∪ X − is the support of X . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
A signed set X is a set X partitionned in two parts ( X + , X − ), where X + is the set of positive elements of X and X − is the set of negatives elements. The set X = X + ∪ X − is the support of X . We say that X is a restriction of Y if and only if X + ⊆ Y + and X − ⊆ Y − . If A is a not signed set and X a signed set then X ∩ A designe the signed set Y with Y + = X + ∩ A et Y − = X − ∩ A . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
The opposite of the set X , denoted by − X , is the signed set defined by ( − X ) + = X − and ( − X ) − = X + . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
The opposite of the set X , denoted by − X , is the signed set defined by ( − X ) + = X − and ( − X ) − = X + . Generally, given a signed set X and a set A we denote by − A X the signed set defined by ( − A X ) + = ( X + \ A ) ∪ ( X − ∩ A ) and ( − A X ) − = ( X − \ A ) ∪ ( X + ∩ A ). We say that the signed set − A X is obtained by an reorientation of A . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
A collection C of signed sets of a finite set E is the set of circuits of a oriented matroid on E if and only if the following axioms are verified : ( C 0) ∅ �∈ C , ( C 1) C = −C , ( C 2) for any X , Y ∈ C , if X ⊆ Y , then X = Y or X = − Y , ( C 3) for any X , Y ∈ C , X � = − Y , and e ∈ X + ∩ Y − , there exists Z ∈ C such that Z + ⊆ ( X + ∪ Y + ) \ { e } and Z − ⊆ ( X − ∪ Y − ) \ { e } . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Observation (a) If sign are not taken into account, ( C 0) , ( C 2) , ( C 3) are reduced to the cicruits axioms of a nonoriented matroid. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Observation (a) If sign are not taken into account, ( C 0) , ( C 2) , ( C 3) are reduced to the cicruits axioms of a nonoriented matroid. (b) All the objects of a matroid M are also consideredas as the objects of the oriented matroid M , in particular the rank of M is the same as the rank of M . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Observation (a) If sign are not taken into account, ( C 0) , ( C 2) , ( C 3) are reduced to the cicruits axioms of a nonoriented matroid. (b) All the objects of a matroid M are also consideredas as the objects of the oriented matroid M , in particular the rank of M is the same as the rank of M . (c) Let M be an oriented matroid E and C the collection of circuits. We clearly have that − A C is the set of circuits of an oriented matroid, detoted by − A M and obtained from M by a reorientation of A . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Observation (a) If sign are not taken into account, ( C 0) , ( C 2) , ( C 3) are reduced to the cicruits axioms of a nonoriented matroid. (b) All the objects of a matroid M are also consideredas as the objects of the oriented matroid M , in particular the rank of M is the same as the rank of M . (c) Let M be an oriented matroid E and C the collection of circuits. We clearly have that − A C is the set of circuits of an oriented matroid, detoted by − A M and obtained from M by a reorientation of A . Notation. We may write X = abcde the signed circuit X defined by X + = { a , d , e } and X − = { b , c } . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Oriented graph Let G be an oriented graph. We obtain the signed circuits from the cycles of G . e f b a c d Then, C = { ( abc ) , ( abd ) , ( aef ) , ( cd ) , ( bcef ) , ( bdef ) , ( abc ) , ( abd ) , ( aef ) , ( cd ) , ( bcef ) , ( bdef ) } . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Vector configuration Let E = { v 1 , . . . , v n } be a set of vectors that generate the space of dimension r over an ordered field. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Vector configuration Let E = { v 1 , . . . , v n } be a set of vectors that generate the space of dimension r over an ordered field. Let us consider a minimal linear dependecy n � λ i v i = 0 i =1 where λ i ∈ I R . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Vector configuration Let E = { v 1 , . . . , v n } be a set of vectors that generate the space of dimension r over an ordered field. Let us consider a minimal linear dependecy n � λ i v i = 0 i =1 where λ i ∈ I R . We obtain an oriented matroid on E by considering the signed sets X = ( X + , X − ) where X + = { i : λ i > 0 } e t X − = { i : λ i < 0 } for all minimal dependencies among the v i . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Let a b c d e f 1 1 0 0 1 0 A = 0 1 1 1 0 0 0 0 0 0 1 1 The columns of A correspond to the following vectors 1 f e c,d 1 a 1 b J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
We can check that the circuits of 1 f e c,d 1 a 1 b are the same as those arising from e f b a c J.L. Ram´ ırez Alfons´ ın d IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity For exemple, ( abc ) correspond to the linear combination
Configurations of points Any configuration of points induce an oriented matroid in the affine space where the signed set of circuits are are the coefficients of minimal affine dependencies of the form � � λ i = 0 , λ i ∈ I λ i v i = 0 with R i i J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
a b c d e f � − 1 � 0 0 3 1 0 A = 0 0 1 0 2 3 J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
a b c d e f � − 1 � 0 0 3 1 0 A = 0 0 1 0 2 3 f e c d a b J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
a b c d e f � − 1 � 0 0 3 1 0 A = 0 0 1 0 2 3 f e c d a b C = { ( abd ) , ( bcf ) , ( def ) , ( ace ) , ( abef ) , ( bcde ) , ( acdf ) , ( abd ) , ( bcf ) , ( def ) , ( ace ) , ( abef ) , ( bcde ) , ( acdf ) } . J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
a b c d e f � − 1 � 0 0 3 1 0 A = 0 0 1 0 2 3 f e c d a b C = { ( abd ) , ( bcf ) , ( def ) , ( ace ) , ( abef ) , ( bcde ) , ( acdf ) , ( abd ) , ( bcf ) , ( def ) , ( ace ) , ( abef ) , ( bcde ) , ( acdf ) } . For instance, circuit ( abd ) correspond to the affine dependecy 3( − 1 , 0) t − 4(0 , 0) t + 1(3 , 0) t = (0 , 0) t with 3 − 4 + 1 = 0. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
The obtained oriented matroid is the one arising from f e c b d a J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
The obtained oriented matroid is the one arising from f e c b d a Geometrically : circuits are minimal Radon partitions. The convex hull of positive elements intersect the convex hull of negatives elements. J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
The obtained oriented matroid is the one arising from f e c b d a Geometrically : circuits are minimal Radon partitions. The convex hull of positive elements intersect the convex hull of negatives elements. For exemple, from circuit ( abd ) we see that point b is in the segment [ a , b ] and from circuit ( abef ) the segment [ a , e ] intersect the segment [ b , f ] J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
Consider the oriented matroid − d M ( A ) obtained by reorienting element d of M ( A ). J.L. Ram´ ırez Alfons´ ın IIMAG, Universit´ e de Montpellier Theory of oriented matroids and convexity
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