van Rees Loops

With Michael Kinyon I wrote a paper investigating latin squares with the maximum possible number of 3x3 subsquares. van Rees showed that you can never have more than n²(n-1)/18 subsquares of order 3 in a latin square of order n. Moreover, he conjectured that this bound is achievable only when n is a power of 3.

In our paper we found a number of algebraically interesting loops/quasigroups whose Cayley tables achieve the van Rees bound. Here they are:

• van Rees loops of order 27

  If you want to know what format this file is in, it is my usual latin squares format. The van Rees loops of order 27 that we found, in the order that they are in the file, are as follows. For each we give a 3-vector G which specifies the orders of the automorphism, autotopism and autoparatopism groups respectively.

  1. The elementary abelian 3-group. G=[11232,8188128,49128768]
  2. The non-abelian 3-group of exponent 3. G=[432,314928,1889568]
  3. A noncommutative, weak inverse property (WIP) loop. G=[216,17496,104976]
  4. A universal left conjugacy closed, left inverse property loop. G=[18,11664,23328]
  5. A commutative WIP loop (isotopic to a Steiner quasigroup that is not the affine one). G=[54,34992,209952]
  6. A CC (conjugacy closed) loop. G=[54,39366,236196]
  7. Another CC loop. G=[108,78732,472392]
  8. A bol loop. G=[72,34992,69984]
  9. A loop with all nuclei trivial (the only such in this list). G=[72,34992,69984]

• a van Rees Steiner quasigroup of order 81 that does not satisfy Marczak's identity.

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