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--- tutorial:apps_matroid [2014/11/05 13:52] – [Constructing a Simple Matroid and Playing Around] schroeter
+++ user_guide:tutorials:apps_matroid [2019/01/25 09:38] – ↷ Page moved from user_guide:apps_matroid to user_guide:tutorials:apps_matroid oroehrig
@@ Line 1: / Line 1: @@
-====== Short Introduction to New Application ''matroid'' ======
+====== Short Introduction to application ''matroid'' ======
-The application ''matroid'' introduced with version 2.9.7 is in its infancy.  This tutorial is meant to show the main features available.
+**This tutorial is also available as a {{ user_guide:apps_matroid.ipynb |jupyter notebook}} for polymake 3.1.**
-To make ''matroid'' your current application start ''polymake'' with the option ''-A matroid'' or use the context switch ''application "matroid";'' from within the ''polymake'' shell.  A permanent setting can be stored with ''set_custom $default_application="matroid";''
+This tutorial is meant to show the main features for handling matroids available. To make ''matroid'' your current application start ''polymake'' with the option ''-A matroid'' or use the context switch
+<code>
+> application "matroid";
+</code>
+from within the ''polymake'' shell.  A permanent setting can be stored with ''%%set_custom $default_application="matroid";%%''
 ===== Constructing a Simple Matroid and Playing Around =====
@@ Line 10: / Line 14: @@
 <code>
-matroid > $M=new Matroid(POINTS=>[[1,0,0],[1,0,1],[1,1,0],[1,0,2]]);
+matroid > $M=new Matroid(VECTORS=>[[1,0,0],[1,0,1],[1,1,0],[1,0,2]]);
 </code>
@@ Line 16: / Line 20: @@
 <code>
-polytope > $M=new matroid::Matroid(POINTS=>[[1,0,0],[1,0,1],[1,1,0],[1,0,2]]);
+polytope > $M=new matroid::Matroid(VECTORS=>[[1,0,0],[1,0,1],[1,1,0],[1,0,2]]);
 </code>
@@ Line 24: / Line 28: @@
 4 3
 </code>
+{{ user_guide:matroid_lattice_of_flats_example.png?nolink&200|}}
-The ''POINTS'' are numbered consecutively, starting from zero.  The bases are encoded as sets of these ordinal numbers.
+The ''VECTORS'' are numbered consecutively, starting from zero.  The bases are encoded as sets of these ordinal numbers.
 <code>
@@ Line 49: / Line 53: @@
 You can also compute other properties, like
 <code>
-matroid > print $M->PAVING, " ", $M->BINARY, " ", new Int( $M->SERIES_PARALLEL ), " ", new Int( $M->CONNECTED );
+matroid > print $M->PAVING?"1":"0", " ",
+matroid (2)> $M->BINARY?"1":"0", " ",
+matroid (3)> $M->SERIES_PARALLEL?"1":"0", " ",
+matroid (4)> $M->CONNECTED?"1":"0";
 1 0 0
@@ Line 62: / Line 69: @@
 Even the lattice of flats could be computed and visualised.
 <code>
-matroid > $lattice=$M->LATTICE_OF_FLATS; foreach (@{$lattice->nodes_of_dim(2)}){print $lattice->FACES->[$_]," "};
+matroid > $lattice=$M->LATTICE_OF_FLATS;
+matroid > foreach (@{$lattice->nodes_of_rank(2)}){print $lattice->FACES->[$_]," "};
 {0 2} {0 1 3} {1 2} {2 3}
@@ Line 73: / Line 81: @@
 matroid > $M->LATTICE_OF_FLATS->VISUAL;
 </code>
-{{ :tutorial:matroid_lattice_of_flats_example.png?nolink&200 |}}
 ===== Matroid Polytopes =====
-You can construct a polytope from the bases of a matroid as the convex hull of the characteristic vectors of the bases.  This is the //matroid polytope// of that matroid, sometimes also called the //matroid bases polytope//.  The matroid polytope of the matroid ''$M'' is a a subobject ''POLYTOPE'' of type ''polytope::Polytope<Rational>''.
+You can construct a polytope from the bases of a matroid as the convex hull of the characteristic vectors of the bases.  This is the //matroid polytope// of that matroid, sometimes also called the //matroid bases polytope//.  The matroid polytope of the matroid ''$M'' is a subobject ''POLYTOPE'' of type ''polytope::Polytope<Rational>''.
 <code>
@@ Line 93: / Line 100: @@
 <code>
-matroid > $C=new Matroid(POINTS=>polytope::cube(3)->VERTICES);
+matroid > $C=new Matroid(VECTORS=>polytope::cube(3)->VERTICES);
 matroid > print $C->N_BASES;
@@ Line 108: / Line 115: @@
 </code>
-Of course you can also construct your matroid from scratch by specifying, e.g., its set of bases or non-bases and then compute other properties. The following constructs the Fano matroid, which is the simplest matroid that cannot be constructed from a vector configuration (over a field with characteristic not equal to two).
+It is also possible to derive a new matroid from others.
+<code>
+matroid > # The arguments are two matroids and for each matroid a basepoint. The basepoints will be identified.
+matroid > $se=series_extension(uniform_matroid(2,3),0,uniform_matroid(1,3),0);
+matroid > print deletion($se,4)->VECTORS;
+0 0
+1 0
+0 1
+1 1
+matroid > $pe=parallel_extension(uniform_matroid(1,3),0,uniform_matroid(2,3),0);
+matroid > print dual(contraction($pe,4))->VECTORS;
+0 0
+1 0
+0 1
+1 1
+matroid > print projective_plane(3)->N_BASES;
+
+matroid > print fano_matroid()->N_BASES;
+
+matroid > print direct_sum(projective_plane(3),fano_matroid())->N_BASES," = 234*28";
+= 234*28
+matroid > print two_sum(uniform_matroid(2,4),0,uniform_matroid(2,4),0)->CIRCUITS;
+{0 1 2}
+{3 4 5}
+{0 1 3 4}
+{0 1 3 5}
+{0 1 4 5}
+{0 2 3 4}
+{0 2 3 5}
+{0 2 4 5}
+{1 2 3 4}
+{1 2 3 5}
+{1 2 4 5}
+</code>
+Of course you can also construct your matroid from scratch by specifying, e.g., its set of bases or non-bases and then compute other properties. The following constructs the Fano matroid, which is the simplest matroid that cannot be constructed from a vector configuration (over a field with a characteristic other than two).
 <code>
-matroid >  $a=new Array<Set<Int> >([0,1,5],[1,2,6],[0,2,3],[1,3,4],[2,4,5],[3,5,6],[0,4,6]);
+matroid > $a=new Array<Set<Int>>([0,1,5],[1,2,6],[0,2,3],[1,3,4],[2,4,5],[3,5,6],[0,4,6]);
 matroid > $m=new Matroid(NON_BASES=>$a,N_ELEMENTS=>7);