If I understand correctly, you are given several sorted lists of numbers $X_1,\ldots, X_n$. The lists are all of the same size $m$, the lists come in a specific enumerated order, and the elements within each list are also ordered (numerically).
I don't know of any existing notation for zipping sets like this, but you could potentially define a notation such as:
$$\bigoplus_{\\{X_1\ldots X_n\\}} \equiv \\{\, \langle X_i[k], X_j[k]\rangle\, : 1\leq i < j \leq n,\; 1\leq k\leq m\\} $$
where here $X_i$ refers to the $i$th set in your specified order, and $X[k]$ refers to the $k$th element of an ordered list.
Then if you had a family $\mathcal{F}$ of unspecified sets, their zipper would be the set $\bigoplus_{\mathcal{F}}$.