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Do you think it should be called bag-power-set rather than bag-power-bag?
set
is replaced by bag
in their names" means that bag-power-bag is the more consistent name. Personally I don't have strong feelings either way and this feels a little like bike-shedding. I'm comfortable with either name.
Duplicates are treated like any other elements: the power bag of
{a, a, b} is {{}, {a}, {a}, {b}, {a, a}, {a, b}, {a, b}, {a, a, b}}.
I'm not sure what there is to clarify: can you explain?