Fix Cayley tables, add operation tables

[rbeezer]

Cayley tables for permutation groups are broken, see #7340.

For other finite algebraic structures, it would be useful for educational purposes to have tables for whatever operation(s) may be present.

Text file included here provides a class that creates a Cayley table object, it can be generalized to provide a similar table for any object with an addition or multiplication - general groups and rings would be the first places to use it.

Depends on #8579.

CC: @jasongrout @dimpase @nthiery

Component: algebra

Keywords: cayley table, operation table

Author: Rob Beezer, Nicolas M. Thiéry

Reviewer: Nicolas M. Thiéry, Jason Grout

Merged: sage-4.4.alpha0

Issue created by migration from https://trac.sagemath.org/ticket/7555

[rbeezer]

Attachment: trac_7555_cayley_table.txt

Experimental Cayley table class, cut/paste into a notebook cell

[rbeezer]

comment:2

Patch contains a new class, OperationTable.

This is meant for educational applications, and is unlikely to be used for anything much bigger than can be displayed on a screen or a sheet of paper. So the speed (it is a bit slow) is not a concern.

Since it only requires a binary operation, I created the groupoids directory, based on the existence of a category by the same name.

Next thing will be an application of this class to generic groups, fixing #7340. There are stubs left here for methods to create color and grayscale graphics of the tables.

[rbeezer]

comment:3

It seems a better place to place this is within the categories framework, making it available automatically in a large number of situations. So it is being reworked right now. The main routine won't change much at all, so time spent reviewing largely won't need to be redone. But you'll want to wait on testing. I'll have an updated patch soon.

[rbeezer]

Attachment: trac_7555_operation_table.patch.gz

[rbeezer]

comment:4

I've tied the new OperationTable class into the categories framework as a multiplication table for Semigroups and as an addition table for Commutative Additive Semigroups. I've also added it as a Cayley table for groups, since I'd like to later expand this somewhat to take advantage of the extra structure in groups.

The old Cayley table was used to build Latin squares. I believe the behavior now with the new cayley table will be identical - IF the identity is the first element of the group.

Had a hard time constructing a nontrivial finite additive semigroup, so the documentation there is barebones right now.

As more structures become available in the categories this should be all ready to go, unchanged. Right now it already makes the multiplication table available for all groups, rather than just permutation groups.