From 4ad619f8d0164d681262de91e882607559bc8636 Mon Sep 17 00:00:00 2001
From: Mohamed Barakat <mohamed.barakat@rwth-aachen.de>
Date: Sat, 17 Oct 2020 17:56:01 +0200
Subject: [PATCH] recode ibmpc..lat1
 examples/DecomposeOnceByRandomEndomorphism.g

---
 examples/DecomposeOnceByRandomEndomorphism.g | 600 +++++++++----------
 1 file changed, 300 insertions(+), 300 deletions(-)

diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g
index add2fb7..12ce041 100644
--- a/examples/DecomposeOnceByRandomEndomorphism.g
+++ b/examples/DecomposeOnceByRandomEndomorphism.g
@@ -1,300 +1,300 @@
-#! @BeginChunk DecomposeOnce
-
-LoadPackage( "CatReps" );
-
-#! @BeginExample
-c3c3 := ConcreteCategoryForCAP( [ [2,3,1], [4,5,6], [,,,5,6,4] ] );
-#! A finite concrete category
-GF3 := HomalgRingOfIntegers( 3 );
-#! GF(3)
-kq := Algebroid( GF3, c3c3 );
-#! Algebroid generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
-SetIsLinearClosureOfACategory( kq, true );
-CatReps := Hom( kq, GF3 );
-#! The category of functors: Algebroid generated by the right quiver
-#! q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices over GF(3)
-d := [[1,1,0,0,0],[0,1,1,0,0],[0,0,1,0,0],[0,0,0,1,1],[0,0,0,0,1]];;
-e := [[0,1,0,0],[0,0,1,0],[0,0,0,0],[0,1,0,1],[0,0,1,0]];;
-f := [[1,1,0,0],[0,1,1,0],[0,0,1,0],[0,0,0,1]];;
-nine := AsObjectInHomCategory( kq, [ 5, 4 ], [ d, e, f ] );
-#! <(1)->5, (2)->4; (a)->5x5, (b)->5x4, (c)->4x4>
-DecomposeOnceByRandomEndomorphism( nine );
-#! fail
-#! @EndExample
-
-#! The above shows that our representation <C>nine</C> is 
-#! indecomposable (with a high probability).
-#! We use the tensor product to generate another representation
-#! <C>fortyone</C>, that is hopefully decomposable, and
-#! inspect the two resulting embeddings <C>iota</C> and
-#! <C>kappa</C>.
-
-#! @BeginExample
-fortyone := TensorProductOnObjects( nine, nine );
-#! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
-result := DecomposeOnceByRandomEndomorphism( fortyone );
-#! [ <(1)->3x25, (2)->1x16>, <(1)->22x25, (2)->15x16> ]
-iota := result[1];
-#! <(1)->3x25, (2)->1x16>
-kappa := result[2];
-#! <(1)->22x25, (2)->15x16>
-Display( fortyone );
-#! An object in The category of functors: Algebroid generated by the
-#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
-#! over GF(3) defined by the following data:
-#! 
-#! 
-#! Image of <(1)>:
-#! A vector space object over GF(3) of dimension 25
-#! 
-#! Image of <(2)>:
-#! A vector space object over GF(3) of dimension 16
-#! 
-#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
-#!  1 1 . . . 1 1 . . . . . . . . . . . . . . . . . .
-#!  . 1 1 . . . 1 1 . . . . . . . . . . . . . . . . .
-#!  . . 1 . . . . 1 . . . . . . . . . . . . . . . . .
-#!  . . . 1 1 . . . 1 1 . . . . . . . . . . . . . . .
-#!  . . . . 1 . . . . 1 . . . . . . . . . . . . . . .
-#!  . . . . . 1 1 . . . 1 1 . . . . . . . . . . . . .
-#!  . . . . . . 1 1 . . . 1 1 . . . . . . . . . . . .
-#!  . . . . . . . 1 . . . . 1 . . . . . . . . . . . .
-#!  . . . . . . . . 1 1 . . . 1 1 . . . . . . . . . .
-#!  . . . . . . . . . 1 . . . . 1 . . . . . . . . . .
-#!  . . . . . . . . . . 1 1 . . . . . . . . . . . . .
-#!  . . . . . . . . . . . 1 1 . . . . . . . . . . . .
-#!  . . . . . . . . . . . . 1 . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . 1 1 . . . . . . . . . .
-#!  . . . . . . . . . . . . . . 1 . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . 1 1 . . . 1 1 . . .
-#!  . . . . . . . . . . . . . . . . 1 1 . . . 1 1 . .
-#!  . . . . . . . . . . . . . . . . . 1 . . . . 1 . .
-#!  . . . . . . . . . . . . . . . . . . 1 1 . . . 1 1
-#!  . . . . . . . . . . . . . . . . . . . 1 . . . . 1
-#!  . . . . . . . . . . . . . . . . . . . . 1 1 . . .
-#!  . . . . . . . . . . . . . . . . . . . . . 1 1 . .
-#!  . . . . . . . . . . . . . . . . . . . . . . 1 . .
-#!  . . . . . . . . . . . . . . . . . . . . . . . 1 1
-#!  . . . . . . . . . . . . . . . . . . . . . . . . 1
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
-#!  . . . . . 1 . . . . . . . . . .
-#!  . . . . . . 1 . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . 1 . 1 . . . . . . . .
-#!  . . . . . . 1 . . . . . . . . .
-#!  . . . . . . . . . 1 . . . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . 1 . 1 . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . 1 . . . . . . . 1 . .
-#!  . . . . . . 1 . . . . . . . 1 .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . 1 . 1 . . . . . 1 . 1
-#!  . . . . . . 1 . . . . . . . 1 .
-#!  . . . . . . . . . 1 . . . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . 1 . 1 . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
-#!  1 1 . . 1 1 . . . . . . . . . .
-#!  . 1 1 . . 1 1 . . . . . . . . .
-#!  . . 1 . . . 1 . . . . . . . . .
-#!  . . . 1 . . . 1 . . . . . . . .
-#!  . . . . 1 1 . . 1 1 . . . . . .
-#!  . . . . . 1 1 . . 1 1 . . . . .
-#!  . . . . . . 1 . . . 1 . . . . .
-#!  . . . . . . . 1 . . . 1 . . . .
-#!  . . . . . . . . 1 1 . . . . . .
-#!  . . . . . . . . . 1 1 . . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#!  . . . . . . . . . . . 1 . . . .
-#!  . . . . . . . . . . . . 1 1 . .
-#!  . . . . . . . . . . . . . 1 1 .
-#!  . . . . . . . . . . . . . . 1 .
-#!  . . . . . . . . . . . . . . . 1
-#! 
-#! A morphism in Category of matrices over GF(3)
-S := DirectSum( [ Source( iota ), Source( kappa ) ] );
-#! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
-Display( S );
-#! An object in The category of functors: Algebroid generated by the
-#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
-#! over GF(3) defined by the following data:
-#! 
-#! 
-#! Image of <(1)>:
-#! A vector space object over GF(3) of dimension 25
-#! 
-#! Image of <(2)>:
-#! A vector space object over GF(3) of dimension 16
-#! 
-#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
-#!  . 2 . . . . . . . . . . . . . . . . . . . . . . .
-#!  1 2 2 . . . . . . . . . . . . . . . . . . . . . .
-#!  . . 1 . . . . . . . . . . . . . . . . . . . . . .
-#!  . . . 1 1 . . . 1 1 . . . . . . . . . . . . . . .
-#!  . . . . 1 1 . . . 1 1 . . . . . . . . . . . . . .
-#!  . . . . . 1 . . . . 1 . . . . . . . . . . . . . .
-#!  . . . . . . 1 1 . . . 1 1 . . . . . . . . . . . .
-#!  . . . . . . . 1 . . . . 1 . . . . . . . . . . . .
-#!  . . . . . . . . 1 1 . . . 1 1 . . . . . . . . . .
-#! . . . . . . . . . 1 1 . . . 1 1 . . . . . . . . .
-#! . . . . . . . . . . 1 . . . . 1 . . . . . . . . .
-#! . . . . . . . . . . . 1 1 . . . 1 1 . . . . . . .
-#! . . . . . . . . . . . . 1 . . . . 1 . . . . . . .
-#! . . . . . . . . . . . . . 1 1 . . . . . . . . . .
-#! . . . . . . . . . . . . . . 1 1 . . . . . . . . .
-#! . . . . . . . . . . . . . . . 1 . . . . . . . . .
-#! . . . . . . . . . . . . . . . . 1 1 . . . . . . .
-#! . . . . . . . . . . . . . . . . . 1 . . . . . . .
-#! . . . . . . . . . . . . . . . . . . 1 1 . 1 1 . .
-#! . . . . . . . . . . . . . . . . . . . 1 1 . 1 1 .
-#! . . . . . . . . . . . . . . . . . . . . 1 . . 1 .
-#! . . . . . . . . . . . . . . . . . . . . . 1 1 . .
-#! . . . . . . . . . . . . . . . . . . . . . . 1 1 .
-#! . . . . . . . . . . . . . . . . . . . . . . . 1 .
-#! . . . . . . . . . . . . . . . . . . . . . . . . 1
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
-#!  2 . . . . . . . . . . . . . . .
-#!  1 . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . 1 . . . . . . . . .
-#!  . . . . . . . 1 . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . 1 . 1 . . . . . . .
-#!  . . . . . . . 1 . . . . . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#!  . . . . . . . . . . . 1 . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . 1 . 1 . . .
-#!  . . . . . . . . . . . 1 . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . 1 . . . . . . . 1 .
-#!  . . . . . . . 1 . . . . . . . 1
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . . . . 1 . . . . .
-#!  . . . . . . . . . . . 1 . . . .
-#!  . . . . . . . . . . . . . . . .
-#!  . . . . . . . 2 . . 1 . 1 . . 2
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
-#!  1 . . . . . . . . . . . . . . .
-#!  . 1 1 . . 1 1 . . . . . . . . .
-#!  . . 1 1 . . 1 1 . . . . . . . .
-#!  . . . 1 . . . 1 . . . . . . . .
-#!  . . . . 1 . . . 1 . . . . . . .
-#!  . . . . . 1 1 . . 1 1 . . . . .
-#!  . . . . . . 1 1 . . 1 1 . . . .
-#!  . . . . . . . 1 . . . 1 . . . .
-#!  . . . . . . . . 1 . . . 1 . . .
-#!  . . . . . . . . . 1 1 . . . . .
-#!  . . . . . . . . . . 1 1 . . . .
-#!  . . . . . . . . . . . 1 . . . .
-#!  . . . . . . . . . . . . 1 . . .
-#!  . . . . . . . . . . . . . 1 1 .
-#!  . . . . . . . . . . . . . . 1 1
-#!  . . . . . . . . . . . . . . . 1
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! @EndExample
-
-#! Comparing the matrices of <C>fortyone</C> with those of 
-#! <C>S</C>, we see
-#! that after decomposing once, we have separated one small
-#! matrix on the diagonal: A $3\times 3$-matrix from <C>S(kq.a)</C>,
-#! a $3 \times 1$-matrix from <C>S(kq.b)</C> and a $1\times 1$-matrix
-#! from <C>S(kq.c)</C>. This matches with the source of the 
-#! embedding <C>iota</C>.
-
-#! @BeginExample
-Display( iota );
-#! A morphism in The category of functors: Algebroid generated by the
-#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
-#! over GF(3) defined by the following data:
-#! 
-#! 
-#! Image of <(1)>:
-#!  2 2 . 1 1 . . . . . . . . . . 1 2 1 2 1 . . . . .
-#!  1 2 1 2 1 1 2 1 2 1 . . . . . 2 . . 1 . 2 . . 1 .
-#!  . . 2 . . . 1 2 . 2 2 . . 1 . . . 1 . . . 2 . . 1
-#! 
-#! A split monomorphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of <(2)>:
-#!  . . . . . . . . . . . . . . . 1
-#! 
-#! A split monomorphism in Category of matrices over GF(3)
-Display( Source( iota) );
-#! An object in The category of functors: Algebroid generated by the
-#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
-#! over GF(3) defined by the following data:
-#!
-#! Image of <(1)>:
-#! A vector space object over GF(3) of dimension 3
-#! 
-#! Image of <(2)>:
-#! A vector space object over GF(3) of dimension 1
-#! 
-#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
-#!  . 2 .
-#!  1 2 2
-#!  . . 1
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
-#!  2
-#!  1
-#!  .
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! 
-#! 
-#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
-#!  1
-#! 
-#! A morphism in Category of matrices over GF(3)
-#! @EndExample
-
-#! We can then look at the other embedding of the direct sum 
-#! decomposition, <C>kappa</C>. The iteration of 
-#! <C>WeakDirectSumDecomposition</C> will continue then
-#! with <C>Source( kappa )</C>. Each time the random
-#! endomorphism will decompose the representation,
-#! lowering the dimensions of each object at most by $3$.
-
-#! @Example
-Source( kappa );
-#! <(1)->22, (2)->15; (a)->22x22, (b)->22x15, (c)->15x15>
-result2 := DecomposeOnceByRandomEndomorphism( Source( kappa ) );
-#! [ <(1)->3x22, (2)->3x15>, <(1)->19x22, (2)->12x15> ]
-#! @EndExample
-#! @EndChunk
+#! @BeginChunk DecomposeOnce
+
+LoadPackage( "CatReps" );
+
+#! @BeginExample
+c3c3 := ConcreteCategoryForCAP( [ [2,3,1], [4,5,6], [,,,5,6,4] ] );
+#! A finite concrete category
+GF3 := HomalgRingOfIntegers( 3 );
+#! GF(3)
+kq := Algebroid( GF3, c3c3 );
+#! Algebroid generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
+SetIsLinearClosureOfACategory( kq, true );
+CatReps := Hom( kq, GF3 );
+#! The category of functors: Algebroid generated by the right quiver
+#! q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices over GF(3)
+d := [[1,1,0,0,0],[0,1,1,0,0],[0,0,1,0,0],[0,0,0,1,1],[0,0,0,0,1]];;
+e := [[0,1,0,0],[0,0,1,0],[0,0,0,0],[0,1,0,1],[0,0,1,0]];;
+f := [[1,1,0,0],[0,1,1,0],[0,0,1,0],[0,0,0,1]];;
+nine := AsObjectInHomCategory( kq, [ 5, 4 ], [ d, e, f ] );
+#! <(1)->5, (2)->4; (a)->5x5, (b)->5x4, (c)->4x4>
+DecomposeOnceByRandomEndomorphism( nine );
+#! fail
+#! @EndExample
+
+#! The above shows that our representation <C>nine</C> is
+#! indecomposable (with a high probability).
+#! We use the tensor product to generate another representation
+#! <C>fortyone</C>, that is hopefully decomposable, and
+#! inspect the two resulting embeddings <C>iota</C> and
+#! <C>kappa</C>.
+
+#! @BeginExample
+fortyone := TensorProductOnObjects( nine, nine );
+#! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
+result := DecomposeOnceByRandomEndomorphism( fortyone );
+#! [ <(1)->3x25, (2)->1x16>, <(1)->22x25, (2)->15x16> ]
+iota := result[1];
+#! <(1)->3x25, (2)->1x16>
+kappa := result[2];
+#! <(1)->22x25, (2)->15x16>
+Display( fortyone );
+#! An object in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 25
+#! 
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 16
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  1 1 . . . 1 1 . . . . . . . . . . . . . . . . . .
+#!  . 1 1 . . . 1 1 . . . . . . . . . . . . . . . . .
+#!  . . 1 . . . . 1 . . . . . . . . . . . . . . . . .
+#!  . . . 1 1 . . . 1 1 . . . . . . . . . . . . . . .
+#!  . . . . 1 . . . . 1 . . . . . . . . . . . . . . .
+#!  . . . . . 1 1 . . . 1 1 . . . . . . . . . . . . .
+#!  . . . . . . 1 1 . . . 1 1 . . . . . . . . . . . .
+#!  . . . . . . . 1 . . . . 1 . . . . . . . . . . . .
+#!  . . . . . . . . 1 1 . . . 1 1 . . . . . . . . . .
+#!  . . . . . . . . . 1 . . . . 1 . . . . . . . . . .
+#!  . . . . . . . . . . 1 1 . . . . . . . . . . . . .
+#!  . . . . . . . . . . . 1 1 . . . . . . . . . . . .
+#!  . . . . . . . . . . . . 1 . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . 1 1 . . . . . . . . . .
+#!  . . . . . . . . . . . . . . 1 . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . 1 1 . . . 1 1 . . .
+#!  . . . . . . . . . . . . . . . . 1 1 . . . 1 1 . .
+#!  . . . . . . . . . . . . . . . . . 1 . . . . 1 . .
+#!  . . . . . . . . . . . . . . . . . . 1 1 . . . 1 1
+#!  . . . . . . . . . . . . . . . . . . . 1 . . . . 1
+#!  . . . . . . . . . . . . . . . . . . . . 1 1 . . .
+#!  . . . . . . . . . . . . . . . . . . . . . 1 1 . .
+#!  . . . . . . . . . . . . . . . . . . . . . . 1 . .
+#!  . . . . . . . . . . . . . . . . . . . . . . . 1 1
+#!  . . . . . . . . . . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  . . . . . 1 . . . . . . . . . .
+#!  . . . . . . 1 . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . 1 . 1 . . . . . . . .
+#!  . . . . . . 1 . . . . . . . . .
+#!  . . . . . . . . . 1 . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . 1 . 1 . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . 1 . . . . . . . 1 . .
+#!  . . . . . . 1 . . . . . . . 1 .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . 1 . 1 . . . . . 1 . 1
+#!  . . . . . . 1 . . . . . . . 1 .
+#!  . . . . . . . . . 1 . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . 1 . 1 . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  1 1 . . 1 1 . . . . . . . . . .
+#!  . 1 1 . . 1 1 . . . . . . . . .
+#!  . . 1 . . . 1 . . . . . . . . .
+#!  . . . 1 . . . 1 . . . . . . . .
+#!  . . . . 1 1 . . 1 1 . . . . . .
+#!  . . . . . 1 1 . . 1 1 . . . . .
+#!  . . . . . . 1 . . . 1 . . . . .
+#!  . . . . . . . 1 . . . 1 . . . .
+#!  . . . . . . . . 1 1 . . . . . .
+#!  . . . . . . . . . 1 1 . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . 1 1 . .
+#!  . . . . . . . . . . . . . 1 1 .
+#!  . . . . . . . . . . . . . . 1 .
+#!  . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+S := DirectSum( [ Source( iota ), Source( kappa ) ] );
+#! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
+Display( S );
+#! An object in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 25
+#! 
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 16
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  . 2 . . . . . . . . . . . . . . . . . . . . . . .
+#!  1 2 2 . . . . . . . . . . . . . . . . . . . . . .
+#!  . . 1 . . . . . . . . . . . . . . . . . . . . . .
+#!  . . . 1 1 . . . 1 1 . . . . . . . . . . . . . . .
+#!  . . . . 1 1 . . . 1 1 . . . . . . . . . . . . . .
+#!  . . . . . 1 . . . . 1 . . . . . . . . . . . . . .
+#!  . . . . . . 1 1 . . . 1 1 . . . . . . . . . . . .
+#!  . . . . . . . 1 . . . . 1 . . . . . . . . . . . .
+#!  . . . . . . . . 1 1 . . . 1 1 . . . . . . . . . .
+#! . . . . . . . . . 1 1 . . . 1 1 . . . . . . . . .
+#! . . . . . . . . . . 1 . . . . 1 . . . . . . . . .
+#! . . . . . . . . . . . 1 1 . . . 1 1 . . . . . . .
+#! . . . . . . . . . . . . 1 . . . . 1 . . . . . . .
+#! . . . . . . . . . . . . . 1 1 . . . . . . . . . .
+#! . . . . . . . . . . . . . . 1 1 . . . . . . . . .
+#! . . . . . . . . . . . . . . . 1 . . . . . . . . .
+#! . . . . . . . . . . . . . . . . 1 1 . . . . . . .
+#! . . . . . . . . . . . . . . . . . 1 . . . . . . .
+#! . . . . . . . . . . . . . . . . . . 1 1 . 1 1 . .
+#! . . . . . . . . . . . . . . . . . . . 1 1 . 1 1 .
+#! . . . . . . . . . . . . . . . . . . . . 1 . . 1 .
+#! . . . . . . . . . . . . . . . . . . . . . 1 1 . .
+#! . . . . . . . . . . . . . . . . . . . . . . 1 1 .
+#! . . . . . . . . . . . . . . . . . . . . . . . 1 .
+#! . . . . . . . . . . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  2 . . . . . . . . . . . . . . .
+#!  1 . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . 1 . . . . . . . . .
+#!  . . . . . . . 1 . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . 1 . 1 . . . . . . .
+#!  . . . . . . . 1 . . . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . 1 . 1 . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . 1 . . . . . . . 1 .
+#!  . . . . . . . 1 . . . . . . . 1
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . 2 . . 1 . 1 . . 2
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  1 . . . . . . . . . . . . . . .
+#!  . 1 1 . . 1 1 . . . . . . . . .
+#!  . . 1 1 . . 1 1 . . . . . . . .
+#!  . . . 1 . . . 1 . . . . . . . .
+#!  . . . . 1 . . . 1 . . . . . . .
+#!  . . . . . 1 1 . . 1 1 . . . . .
+#!  . . . . . . 1 1 . . 1 1 . . . .
+#!  . . . . . . . 1 . . . 1 . . . .
+#!  . . . . . . . . 1 . . . 1 . . .
+#!  . . . . . . . . . 1 1 . . . . .
+#!  . . . . . . . . . . 1 1 . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . 1 . . .
+#!  . . . . . . . . . . . . . 1 1 .
+#!  . . . . . . . . . . . . . . 1 1
+#!  . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! @EndExample
+
+#! Comparing the matrices of <C>fortyone</C> with those of 
+#! <C>S</C>, we see
+#! that after decomposing once, we have separated one small
+#! matrix on the diagonal: A $3\times 3$-matrix from <C>S(kq.a)</C>,
+#! a $3 \times 1$-matrix from <C>S(kq.b)</C> and a $1\times 1$-matrix
+#! from <C>S(kq.c)</C>. This matches with the source of the 
+#! embedding <C>iota</C>.
+
+#! @BeginExample
+Display( iota );
+#! A morphism in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of <(1)>:
+#!  2 2 . 1 1 . . . . . . . . . . 1 2 1 2 1 . . . . .
+#!  1 2 1 2 1 1 2 1 2 1 . . . . . 2 . . 1 . 2 . . 1 .
+#!  . . 2 . . . 1 2 . 2 2 . . 1 . . . 1 . . . 2 . . 1
+#! 
+#! A split monomorphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of <(2)>:
+#!  . . . . . . . . . . . . . . . 1
+#! 
+#! A split monomorphism in Category of matrices over GF(3)
+Display( Source( iota) );
+#! An object in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#!
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 3
+#! 
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 1
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  . 2 .
+#!  1 2 2
+#!  . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  2
+#!  1
+#!  .
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! @EndExample
+
+#! We can then look at the other embedding of the direct sum 
+#! decomposition, <C>kappa</C>. The iteration of 
+#! <C>WeakDirectSumDecomposition</C> will continue then
+#! with <C>Source( kappa )</C>. Each time the random
+#! endomorphism will decompose the representation,
+#! lowering the dimensions of each object at most by $3$.
+
+#! @Example
+Source( kappa );
+#! <(1)->22, (2)->15; (a)->22x22, (b)->22x15, (c)->15x15>
+result2 := DecomposeOnceByRandomEndomorphism( Source( kappa ) );
+#! [ <(1)->3x22, (2)->3x15>, <(1)->19x22, (2)->12x15> ]
+#! @EndExample
+#! @EndChunk