//European Example. The hierarchy of blocks.
//Please view in a non-proportional font such as Lucida Console. 

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THE BLOCK SYSTEM FOR LEVEL 1.
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This block system is obtained from the 596 legal permutations for level 1 listed in file EELevel1Permutations.txt
It contains a total of 14 blocks. The reader can verify that this block system is invariant for all 596 permutations. 

L1 BLOCKS:   (2 8)(3 7)(0  6 10)(4 5)(1 9)(11 13 14)(12 15 16)(17)(18 19 22)(21 20 23)(24)(25 26 28)(30 27)(29 31 32)
L1 RENAME:     A    B      C      D    E       F         G      H      I         J      K      L        M      N
ORDER          T    T      T      N    T       N         N      -      N         N      -      N        N      T

The blocks in the system have been assigned names for the purpose of reference, as indicated. 
These blocks become the elements of the causal set for level 2.
The row "ORDER" is explained below. 

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THE INDUCTION AND ENCAPSULATION OF PRECEDENCE RELATIONS FROM LEVEL 1 INTO THE BLOCK SYSTEM OF LEVEL 1
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For the purpose of generating the causal set for level 2, it is necessary to calculate the precedence relations for 
the blocks of level 1. They are obtained by induction (resulting in inter-block relations) or encapsulation (resulting 
in intra-block relations) of the precedence relations from level 1. Only the induced relations are indicated below. 

Reverse for level 1 elements/blocks.
0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32 
C  E  A  B  D  D  C  B  A  E   C   F   G   F   F   G   G   H   I   I   J   J   I   J   K   L   L   M   L   N   M   N   N 

  PRECEDENCE RELATIONS          INTER-BLOCK        INTRA-BLOCK RELATIONS
  IN LEVEL 1			RELATIONS          (ENCAPSULATED)
  0  <   4    5    6   10	C  <  D            0 < 6    0 < 10
  1  <   9   14			E  <  F            1 < 9
  2  <   8   10			A  <  C            2 < 8
  3  <   7   10			B  <  C            3 < 7
  4  <   9   13			D  <  E   F        -
  5  <   9   11			D  <  E   F        -
  6  <  10			-                  6 < 10
  7  <  10			B  <  C            -
  8  <  10			A  <  C            -
  9  <  11   13   14		E  <  F            -
 10  <  11   13   14		C  <  F            -
 11  <  12			F  <  G            -
 12  <  17   18			G  <  H   I        -
 13  <  15			F  <  G            -
 14  <  16  			F  <  G            -
 15  <  17   19			G  <  H   I        -
 16  <  17   22			G  <  H   I        -
 17  <  18   19   22		H  <  I            -
 18  <  20			I  <  J            -
 19  <  21			I  <  J            -
 20  <  24   25			J  <  K   L        -
 21  <  24   28			J  <  K   L        -
 22  <  23			I  <  J            -
 23  <  24   26			J  <  K   L        -
 24  <  25   26   28		K  <  L            -
 25  <  27			L  <  M            -
 26  <  29			L  <  N            -
 27  <  31			M  <  N            -
 28  <  30			L  <  M            -
 29  <  31   32			-                  29 < 31   29 < 32
 30  <  31			M  <  N            -
 31  <  32			-                  31 < 32

Some inter-block relations are induced multiple times. 
There are 18 independent inter-block precedence relations. They are: 
{A<C, B<C, C<D, C<F, D<E, D<F, E<F, F<G, G<H, G<I, H<I, I<J, J<K, J<L, K<L, L<M, L<N, M<N}.

There are a total of 9 encapsulated precedence relations. They are:
{0<6, 0<10, 1<9, 2<8, 3<7, 6<10, 29<31, 29<32, 31<32}.

The encapsulated relations indicate the existence of some type of order in some blocks. 
For example, blocks C(0,6,10) and N(29,31,32) are totally ordered.
This is indicated in the row labelled "ORDER" in th table above, as follows:
T=TOTAL ORDER     P=PARTIAL ORDER   N=NO ORDER.

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THE CAUSAL SET FOR LEVEL 2 AND THE BLOCK SYSTEM FOR LEVEL 2
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The causal set for level 2 has 14 elements and 18 precedence relations among them. This set is:
S2 = {A,B,C,D,E,F,G,H,I,J,K,L,M,N}
O2 = {A<C,B<C,C<D,C<F,D<E,D<F,E<F,F<G,G<H,G<I,H<I,I<J,J<K,J<L,K<L,L<M,L<N,M<N}

The symmetry of this causal set (S2,O2) is represented by only two legal permutations:
A B C D E F G H I J K L M N
B A C D E F G H I J K L M N

The corresponding block system for level 2 is:
L2 BLOCKS:  (A B)(C D E F G H I J K L M N)
L2 RENAME:    P              Q

There is only one induced relation: P < Q.

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LEVELS 3 AND 4
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The causal set for level 3 is: 
S3 = {P,Q}
O3 = {P<Q}
which has only 1 block: (P,Q), and no precedence relations.

This block is the only element of level 4.

In summary, the hierarchy for the European Example has 4 levels total. 
  Level 1: 33 elements, 55 precedence relations.
  Level 2: 14 elements, 18 precedence relations.
  Level 3:  2 elements,  1 precedence relation.
  Level 4:  1 element,   0 precedence relations. 
The compression of the original information all the way down to 1 single elements is evident. 
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