[loops] 8 Libraries of small loops

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# 8 Libraries of small loops

### Sections

1. A typical library
2. Left Bol loops
3. Small Moufang loops
4. Steiner loops
5. CC-loops
6. Paige loops
7. Interesting loops

labellib Libraries of small loops form an integral part of LOOPS. We describe them here.

## 8.1 A typical library

A library named my Library is stored in file `data/mylibrary.tbl`, and the corresponding data structure is named `my_library_data`.

The array `my_library_data` consists of three lists

- `my_library_data[ 1 ]` is a list of orders for which there is at least one loop in the library,
- `my_library_data[ 2 ][ k ]` is the number of loops of order `my_library_data[ 1 ][ k ]` in the library,
- `my_library_data[ 3 ][ s ]` contains data necessary to produce the sth loop in the library.

The format of `my_library_data[ 3 ]` depends on the particular library and is not standardized in any way.

The user can retrieve the mth loop of order n from library named my Library according to the template

- `MyLibraryLoop(` n`,` m`) F`

  It is also possible to obtain the same loop with

- `LibraryLoop(` name`,` n`,` m`) F`

  where name is the name of the library.

  For example, when the library is called left Bol, the corresponding data file is called `data/leftbol.tbl`, the corresponding data structure is named `left_bol_data`, and the mth left Bol loop of order n is obtained via

  `LeftBolLoop(` n`,` m`)`

  or via

  `LibraryLoop("left Bol",` n`,` m`)`

  We are now going to describe the individual libraries in detail. A brief information about the library named name can also be obtained in LOOPS with

- `DisplayLibraryInfo(` name`) F`

  ## 8.2 Left Bol loops

  The library named left Bol contains all 6 nonassociative left Bol loops of order 8. Following the general pattern, the mth nonassociative left Bol loop of order n is obtained by

- `LeftBolLoop(` n`,` m`) F`

  We intend to enlarge this library significantly in future versions of LOOPS, when the classification of small Bol loops is completed.

  ## 8.3 Small Moufang loops

  The library named Moufang contains all nonassociative Moufang loops of order less than 64, and additional 4262 nonassociative Moufang loops of order 64. It is possible that there are no other nonassociative Moufang loops of order 64 than those contained in the library.

  The mth nonassociative Moufang loop of order n is obtained by

- `MoufangLoop(` n`,` m`) F`

  For n ≤ 63, our catalog numbers coincide with those of Goodaire et al. Goodaire.

  The extent of the library is summarized below:

  order
  12
  16
  20
  24
  28
  32
  36
  40
  42
  44
  48
  52
  54
  56
  60
  64
  loops in the libary
  1
  5
  1
  5
  1
  71
  4
  5
  1
  1
  51
  1
  2
  4
  5
  4262

  The octonion loop multiplication loop of the ± basis elements in the 8-dimensional standard real octonion algebra) is `MoufangLoop( 16, 3 )`.

  Since we would like to know if there are additional nonassociative Moufang loops of order 64, we have implemented the function

- `IsomorphismTypeOfMoufangLoop(` L`) F`

  If L is a Moufang loop cataloged in LOOPS as the mth Moufang loop of order n, the function returns [[n,m],p], where p is a permutation of [1,...,n] that is an isomorphism from L to the cataloged copy of L. If n=64 and L is Moufang loop not cataloged in LOOPS, the user is prompted to contact the authors of LOOPS.

  In order to speed up the function `IsomorphismTypeOfMoufangLoop`, we have precalculated and stored in the data file `data\moufang_discriminators.tbl` the discriminators of all Moufang loops in the library. The file is rather large (850 KB), and took about 20 minutes to precalculate. You can delete the file if you will not use `IsomorphismTypeOfMoufangLoop`.

      gap> D := DirectProduct( MoufangLoop( 16, 2 ), CyclicGroup( 2 ) );
      <loop of order 32>
      gap> IsomorphismTypeOfMoufangLoop( D );
      [ [ 32, 2 ], (2,3,12,20,11,29,23,13,30,31,28,27,22,15,32,18,10,19,16,24,14,
          25,21,8,7,6,9,17,5) ]

  ## 8.4 Steiner loops

  Here is how the libary Steiner is described within LOOPS:

      gap> DisplayLibraryInfo( "Steiner" );
      The library contains all nonassociative Steiner loops of order less or equal to 16.
      It also contains the associative Steiner loops of order 4 and 8.
      ------
      Extent of the library:
         1 loop of order 4
         1 loop of order 8
         1 loop of order 10
         2 loops of order 14
         80 loops of order 16
      true

  The mth Steiner loop of order n is obtained by

- `SteinerLoop(` n`,` m`) F`

  Our catalog numbers coincide with those of Colbourn and Rosa CR.

  ## 8.5 CC-loops

  By results of Kunen Kunen, for every odd prime p there are precisely 3 nonassociative conjugacy closed loopsindexconjugacy closed loop of order p2. Csörgö and Drápal CD described these 3 loops by multiplicative formulas on Zp2 and Zp ×Zp.

  Case m = 1: Let k be the smallest positive integer relatively prime to p and such that k is a square modulo p (i.e., k=1). Define multiplication on Zp2 by x·y = x + y + kpx2y.

  Case m = 2: Let k be the smallest positive integer relatively prime to p and such that k is not a square modulo p. Define multiplication on Zp2 by x·y = x + y + kpx2y.

  Case m = 3: Define multiplication on Zp ×Zp by (x,a)(y,b) = (x+y, a+b+x2y ).

  Moreover, Wilson Wilson constructed a nonassociative CC-loop of order 2p for every odd prime p, and Kunen Kunen showed that there are no other nonassociative CC-loops of this order. Here is the construction:

  Let N be an additive cyclic group of order n > 2, N = 〈1〉. Let G be the additive cyclic group of order 2. Define multiplication on L = G ×N as follows:

  (0,m)(0,n) = ( 0, m + n ),
  (0,m)(1,n) = ( 1, −m + n ),
  (1,m)(0,n) = ( 1, m + n ),
  (1,m)(1,n) = ( 0, 1 − m + n )·

  The CC-loops described above can be obtained by

- `CCLoop(` n`,` m`) F`

  ## 8.6 Paige loops

  Paige loopsindexPaige loop are nonassociative finite simple Moufang loops. By Li, there is precisely one Paige loop for every finite field GF(q).

  The library named Paige contains the smallest nonassociative simple Moufang loop

- `PaigeLoop(` 2`) F`

  ## 8.7 Interesting loops

  The library named interesting contains some loops that are illustrative for the theory of loops. At this point, the library contains a nonassociative loop of order 5, a nonassociative nilpotent loop of order 6, a nonMoufang left Bol loop of order 16, and the loop of sedenionsindexsedenions of order 32 (sedenions generalize octonions).

  The loops are obtained with

- `InterestingLoop(` n`,` m`) F`

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