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Interesting cases would be

$A=[1,1], B=[2]$

$[(a,b) \mid a \in A \wedge b \in B] = [(1,2),(1,2)]$ ?



$[x \mid c \in C \wedge x = c \mod 2] = [1,0,1]$ ?

The only kind of informal definition for multiset-builder notation I could find was a part in the preliminaries section of A Potpourri of Reason Maintenance Methods.

Are there other (more rigorous) studies on that topic?

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Unless you're developing multiset-theoretic foundations for mathematics, you don't really need a formal theory: you just a syntax and how to interpret it. – Hurkyl Feb 23 '13 at 15:30
Right, it just came to my mind that the heading is a bit unfortunate, and the theory is just multiset theory with notations on top of that. Edit: I changed the heading. – letmaik Feb 23 '13 at 15:31

I'd like to describe a notation I used myself a bit. It's really straight-forward and is quite useful for certain cases.

$A = \{1,2,3\}, B=[1,1,2]$

$[x_{[y]} \mid x \in A \wedge y=2x] = [1,1,2,2,2,2,3,3,3,3,3,3]$

$[x_{[2n]} \mid x \in^n B] = [1,1,1,1,2,2]$

So, in general, $[x_{[y]} \mid P(x,y)]$ is the multiset of all elements $x$ with multiplicity $y$ that satisfy the predicate $P(x,y)$.

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