BasicNotions.hpp File Reference

Plans regarding the basic notions about latin squares. More...

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Detailed Description

Plans regarding the basic notions about latin squares.

The notion of a "latin square"
Properties of latin squares
  • Regarding considering a latin square L as a quasigroup Q (via ls2qgrp), we have the following basic properties:
    1. Q can be unital (this is automatically given when L is reduced.
    2. Q can be a associative (that is, a group; this includes unitality).
    3. Q can be commutative (that is, L is symmetric).
  • If we consider a latin square L over a set H together with matrix isomorphisms (which independently can permute rows and columns, but leaving values fixed --- a principal isotopy):
    1. L is always isomorphic to a reduced latin square, and thus (if not empty) yields a unital quasigroup.
    2. For the other properties of quasigroups (which only arise once we fix a (general) matrix isomorphism to an observed square matrix), associativity (yielding a group) and commutativity (yielding a commutative quasigroup), it seems unlikely that they can always be realised.
    3. And combination of commutativity with unitality might again give a new property (yielding a commutative unital quasigroup).
    4. Whether L can yield a commutative quasigroup is exactly whether L is self-polar (see ComputerAlgebra/CombinatorialMatrices/Lisp/Isomorphisms.mac); also the weaker notion of self-duality seems of interest here.
    5. As
      shows, the latin square nassoc_rls is not even isomorphic to its transposed (as a general matrix).
Enumeration and classification
  • Since the isomorphism types given by square matrices are handled in module Quasigroups (see "Enumeration and classification" in ComputerAlgebra/Algebra/Lisp/Groupoids/Quasigroups/plans/general.hpp), here we should (only) consider the isomorphism types given by general isomorphism for matrices (that is, principal isotopy).
  • Then we need to count (total and non-isomorphic) the number of such basic types of latin squares which (can) yield unital, associative etc. latin squares (see above).
  • Interesting also, how a genereral isomorphism type is sub-divided into the special isomorphism types (for square matrices).
  • Is the general-matrix-isomorphism already handled when we consider only reduced matrices? But what would then be the notion of isomorphism between fully standardised squares?
  • It would be good if all reduced latin squares of order n could be enumerated for small n. Then a general construction yields the enumeration of all latin squares of order n.
    1. Easiest to just list one (representative) element of each paratopy class.
    2. The number of paratopy classes is given by sequence A003090:
      1, 1, 1, 2, 2, 12, 147, 283657, 19270853541, 34817397894749939
    3. While the number of isotopy classes is given by sequence A040082:
      1, 1, 1, 2, 2, 22, 564, 1676267, 115618721533, 208904371354363006
    4. The simplest latin square is the cyclic group of order n; for 1 <= n <= 3 this is the only paratopy class, and in fact the only isotopy class.
    5. For n <= 4 every loop is a group. So the second isotopy class for n=4 is given by the Kleinian 4-group.
    6. While the second isotopy class for n=5 is given by nassoc_rls.
    7. Definitely also for n=6 we should have representatives of the 12 paratopy classes and the 22 isotopy classes in our library.
  • DONE The knowledge on counting reduced latin squares should be made available, say num_rls(n).
  • This is sequence A000315:
    1, 1, 1, 4, 56, 9408, 16942080, 535281401856, 377597570964258816, 
  • DONE While for the number of arbitrary latin squares (with value set {1,...,n}) we have num_ls(n) = num_rls(n) * n! * (n-1)!.
  • It would perhaps be better if for example we had
    num_ls(12) = 19120211066880000 * num_rls(12)
    (instead of "19120211066880000*unknown")?
  • See above for isotopy classes and paratopy classes.

Definition in file BasicNotions.hpp.