Which definitions of builder notation exist for multiset theory?

Question

Interesting cases would be

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

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

or

$C=[1,2,3]$

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

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?