My initial thoughts about group domains related to homotopy in FriCAS is
that there is a need for at least 4 group domains shown at each corner
of this square:
PermutationGroup <-----equivalent-----> GroupPresentation
| if finite |
| |
contains set of contains set of
| |
V V
Permutation <-------------------------------> Word
The domains at the bottom of the diagram are implementations of Group
category. So in these cases % will contain something representing a
single element of the group such as a single permutation or a single
word. Functions will be from Group such as '*'.
The domains at the top of the diagram have % which holds information
about the whole group so it identifies it as say 'cyclic group 5' or
'dihedral group 3'. The functions will be functions on the whole group
such as: sum, product, quotient, subgroup, order, orbit, etc.
(I don't think Bill likes this way of describing it? I think the
distinction is valid but can you think of a more mathematical way to
describe the distinction? Perhaps in terms of initial and terminal
algebras?)
So how does FinitelyPresentedGroup fit in this? It seems to me that
FinitelyPresentedGroup is of type: Type whereas GroupPresentation is of
a specific type:
(1) -> F:=FPG([x,y,z],[])
(1) FinitelyPresentedGroup([x,y,z],[])
Type: Type
(2) -> F2 := groupPresentation([1,2,3],[])
(2) <a b c | >
Type: GroupPresentation
Given this, is it possible to construct functions like sum, product,
quotient, subgroup, order, orbit, etc. on something of type: Type?
If it is possible to do this, why is PermutationGroup not constructed
this way?
Martin B
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