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Dear Dima Pasechnik,

It seems still unclear. If g is a permutation of a set X of

whatever nature, S a subset of X, then S^g is a set, otherwise one

would expect a message like "Error, <set> must be a set in

...OnSets". That is, S^g is uniquely defined. That is , S^g

should be a set, so the appropriate flag of it is 'yes'.

sorry, now I am I bit confused. You were asking about sets in

general and in particular, why 'AddSet' has some problems with sets of

mutable objects and insertion. I think we now all more or less agree

how to solve these kinds of problems. But 'OnSets' does not use

'AddSet', it will apply <g> to all elements of <S> and then sort the

resulting list. Again for the same reasons that 'AddSet' cannot flag

this list as set, if X contains mutable objects, 'OnSets' will not do

it either. Surely, we can do without the 'IsSet' test but again we

decided to play it safe in order to avoid traps for others, even if

this means that GAP will be slower than possible without these tests.

If however your set X contains unchangeable objects like integers,

yes, one could try a bit harder to keep the information that 'OnSets'

returns a set. It is not clear to what amount this will speed up

things, because the test requires only |<S>|-1 comparisons while the

sort needs something like O(|<S>|*log|<S>|) and one has to check if

the image of <s> in <S> under <g> is again unchangeable. In the very

special case of a list/set of integers <S> and permutation <g> one

could do without tests. I will try to do this in the next version but

the speedup - I guess - will not be dramatical, presumably much

smaller than a factor of 2.

I hope we all now understand the problems with sets, but if there are

any further questions, I think we should discuss them in

"gap-trouble". So if anyone wants more information about the internal

structure of sets, simply send a mail to

"gap-trouble@math.rwth-aachen.de"

(there is no need to subscribe to it). If any new insight will come

out of further discussion, we will report this to the gap-forum.

best wishes

Frank Celler

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