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Rename minimal_generating_set
for groups to e.g. minimum_generating_set
or minimal_size_generating_set
?
#4181
Comments
I am opposed to this: |
Do these other examples that come to your mind return a minimal generating set w.r.t. cardinality or w.r.t. subset-relation? |
w.r.t. cardinality -- it need not be contained in the input set of generators, it only needs to generate the same ideal. |
in this case I think that @fingolfin 's proposal to change the name (with deprecations) would extend to these versions as well. In the precise mathematical terms I learned (and it seems @fingolfin as well), |
I am not a group theorist by training, but in the group theory papers that I encountered a minimal generating set always meant a generating set which is minimal in the sense of subset-relation. Edit: Also in the title it mentions only the method for groups. |
In the cases mentioned by Anne we have Nakayama's lemma which implies that a set of generators is minimal w.r.t. subset-relation iff it is minimal w.r.t. cardinality. In the group case, as far as I understand, we think of minimal w.r.t. inclusion. We can then speak of the minimal cardinality of a minimal generating set and for some classes of groups such sets can be reasonably computed. So one question here is what the GAP command refers to: Computing a minimal generating set or computing a |
The function So e.g. if I take the additive group generated by 2 and 3, this is But I think this is contrary to the usual interpretation of "minimal" which is "minimal by inclusion (and similar for "maximal"). That is, for some kind of object "FOO" we usually mean by "$A$ is a minimal FOO" that Anyway, I bring this up because on the GAP side this has frequently lead to confusion in the past when people expected that "minimal generating set" means "minimal with respect to inclusion" (which is much cheaper than finding a minimum generating set) |
Just talked with @wdecker and we agreed that |
I think for groups in the literature this is called "minimum generating set". So I would throw |
Since the question is about avoiding misunderstandings, I do not think that there is a general consensus in the literature about the subtle difference between "minimum generating set" and "minimal generating set". As far as I see, the papers dealing with this topic define the notion they use. (I would call a generating set that is minimal w.r.t. taking subsets irredundant.) |
... for some more clarify on what it does (though we should retain the old name, too, for backwards compatibility)
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