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### 3 Development History

#### 3.1 Versions of the package

The first version of the package, written for GAP 3, formed part of Anne Heyworth's thesis [Hey99] in 1999, but was not made generally available.

Version 0.91 was prepared to run under GAP 4.4.6, in July 2005.

Version 0.94 differed in two significant ways.

• The manual was produced using the GAPDoc package.

• The test file kan/tst/kan_manual.tst set the AssertionLevel to 0 to avoid recursion in the Automata package.

Version 0.95, of 9th October 2007, just fixed file protections and added a CHANGES file.

Version 0.96 was required because the Kan website moved with the rest of the Mathematics website at Bangor.

Version 0.97, of November 18th 2008, deleted temporary fixes which were no longer needed once version 1.12 of Automata became available.

Version 1.01, of August 2011, included minor changes required for GAP 4.5. In particular, the test file kan_manual.tst was replaced by the pair fulltest.tst and parttest.tst.

Version 1.11, of December 2014 was required when the Kan website moved yet again. At the same time a bitbucket repository for the package was started.

Kan became an accepted GAP package in May 2015.

Version 1.28, of May 2017, saw a great many changes to the examples, with the various rewriting systems used to perform reduction of words to reduced form.

#### 3.2 What needs doing next?

There are too many items to list here, but some of the most important are as follows.

• Implement iterators and enumerators for double cosets.

• At present the methods for DoubleCosetsNC and RightCosetsNC in this package return automata, rather than lists of cosets or coset enumerators. This needs to be fixed.

• Provide methods for operations such as DoubleCosetRepsAndSizes.

• Convert the rest of the original GAP 3 version of Kan to GAP 4.

##### 3.2-1 DoubleCosetsAutomaton
 ‣ DoubleCosetsAutomaton( G, U, V ) ( operation )
 ‣ RightCosetsAutomaton( G, V ) ( operation )

Alternative methods for DoubleCosetsNC(G,U,V) and RightCosetsNC(G,V) should be provided in the cases where the group G has a rewriting system or is known to be infinite. At present the functions RightCosetsAutomaton and DoubleCosetsAutomaton return minimized automata, and Iterators for these are not yet available.


gap> F := FreeGroup(2);;
gap> rels := [ F.2^2, (F.1*F.2)^2 ];;
gap> G5 := F/rels;;
gap> genG5 := GeneratorsOfGroup( G5 );;
gap> a := genG5[1];  b := genG5[2];;
gap> U := Subgroup( G5, [a^2] );;
gap> V := Subgroup( G5, [b] );;
gap> L := [2,1,4,3];;
gap> rws5 := ReducedConfluentRewritingSystem( G5, L, "shortlex", 0, "aAbB" );;
gap> dc5 := DoubleCosetsAutomaton( G5, U, V );;
gap> Print( dc5 );
Automaton("det",5,"HKAaBb",[ [ 2, 2, 2, 5, 2 ], [ 2, 2, 1, 2, 1 ], [ 2, 2, 2, \
2, 3 ], [ 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2 ] ],[ 4 ],[ 1 ])\
;;
gap> rc5 := RightCosetsAutomaton( G5, V );;
gap> Print( rc5 );
Automaton("det",6,"HKAaBb",[ [ 2, 2, 2, 6, 2, 2 ], [ 2, 2, 1, 2, 1, 1 ], [ 2, \
2, 3, 2, 2, 3 ], [ 2, 2, 2, 2, 5, 5 ], [ 2, 2, 2, 2, 2, 2 ], [ 2, 2, 2, 2, 2, \
2 ] ],[ 4 ],[ 1 ]);;


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