Improved direct products, with new supporting methods

[wilfwilson]

This PR includes several related changes. Most importantly:

• this PR introduces a new method for DirectProductOp, for a list of either transformation semigroups, or bipartition semigroups, or partial perm semigroups.

Previously, DirectProductOp methods existed for:

• a list of transformation monoids
• a list of transformation monoids-as-semigroups
• an arbitrary list of transformation semigroups
• a list of partial perm inverse monoids

This PR extends this functionality to support all partial perm semigroups, and all bipartition semigroups. It does this with a unified function. Notably, this function replaces the previous method for arbitrary finite transformation semigroups, which used a brute force search to find a generating set for the direct product.

My new method sometimes returns a semigroup with a larger generating set than the previous method, but it finds the answer far faster. To see an example of improved performance, try calculating the direct product of the singular transformation semigroup of degree 4 with itself. This takes 40ms with the new code, and gives a semigroup with 96 generators. The old code takes 4800ms, although it does only use 53 generators. Try a higher degree and the time difference is far starker. The generators returned by the new method can be reduced by SmallGeneratingSet, or similar, if desired.

I have ideas for reducing the number of generators returned my method, but they're still only ideas, and in any case, they would require my (future) pull request concerning the partial order of L- and R-classes of a semigroup, and in particular, maximal L- and R-classes.

The new direct product code is reliant on the new methods called:

• NonTrivialFactorization.

Also introduced are the related new functions:

• IsDecomposableSemigroup (i.e. is S^2 = S?)
• IndecomposableElements (i.e. those elements in S \ S^2)

The indecomposable elements of S are precisely those elements in GeneratorsOfSemigroup(S) without a non-trivial factorization in the semigroup.

[wilfwilson]

This is almost ready for merging, I just need to complete my proof that the algorithm is correct.

[wilfwilson]

It took far longer to prove than I thought it would, but I'm now confident that this PR is mathematically sound, and therefore ready to merge. I'd be grateful for reviews!

[wilfwilson]

@mtorpey and I are both using up-to-date master versions of GAP, libsemigroups, and Semigroups.

[james-d-mitchell]

Ok, so this also happens when the Semigroups package is not loaded at all

[james-d-mitchell]

So, it seems to be an issue in GAP itself, if the first group of tests is commented out.

[james-d-mitchell]

If the contents of the test file bug.tst is:

gap> START_TEST("bug.tst");
gap> IsAbelian("A");
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `IsCommutative' on 1 arguments
gap> Unbind(M);
gap> STOP_TEST("bug.tst");

and I load GAP without Semigroups (or anything else loaded using gap -A), then do

    i := 0;; repeat i := i + 1; until not Test("bug.tst");

then I get, when i is 1665

########> Diff in bug.tst:2
# Input is:
IsAbelian("A");
# Expected output:
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 1st choice method found for `IsCommutative' on 1 arguments
# But found:
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, recursion depth trap (5000)

[james-d-mitchell]

Just going to check if this still holds when the Unbind(M); is removed. Yes, this occurs when i is 1665 for me without the Unbind(M); too.

[james-d-mitchell]

Can @wilfwilson, @flsmith and/or @mtorpey confirm?

[james-d-mitchell]

I'm going to file an issue

[flsmith]

@james-d-mitchell I get the exact same behaviour from that bug.tst

[james-d-mitchell]

Can you please add some tests for Embedding and Projection? Sorry forget it, I see there are plenty of tests.

[james-d-mitchell]

Thanks @flsmith

[wilfwilson]

@james-d-mitchell I got the same behaviour from bug.tst also.

[wilfwilson]

@james-d-mitchell Embedding and Projection are tested extensively by the helper function ProductCheck, which is defined at the top of semidp.tst and used about 30 times throughout the file.

I can add some more visible tests though.

[james-d-mitchell]

Sorry @wilfwilson I didn't see the tests because github collapsed the entire file. I see it is well tested! I'll have a look through, then hopefully we'll get the CI to work, then merge.

[wilfwilson]

Just saw your updated message @james-d-mitchell - I'll only add the test that I'm already working on now.

[wilfwilson]

@james-d-mitchell I'm pleased to report that Travis now passes, thanks to @ChrisJefferson's fix.

[james-d-mitchell]

Awesome, I'll look this over again today.

[james-d-mitchell]

Looks good apart from the minor change to the doc.

[james-d-mitchell]

Also if they are transformation semigroups, no? Also remove "either" since there are more than two possibilities, and also remove the first "or"

[wilfwilson]

It's true that if the args are all transformation semigroups, then the result is a transformation semigroup. But that falls under the next sentence: Otherwise, <C>DirectProduct</C> returns a transformation semigroup.

[james-d-mitchell]

Right you are, I'll merge this just now