  Studying the problem of quasicommuting quantum minors, Leclerc and Zelevinsky
introduced in 1998 the notion of weakly separated sets in $[n]:=\{1,\ldots,
n\}$. Moreover, they raised several conjectures on the purity for this
symmetric relation, in particular, on the Boolean cube $2^{[n]}$. In
0909.1423[math.CO] we proved these purity conjectures for the Boolean cube
$2^{[n]}$, the discrete Grassmanian ${[n]\choose r}$, and some other
set-systems. Oh, Postnikov, and Speyer in arxiv:1109.4434 proved the purity for
weakly separated collections inside a positroid which contain a Grassmann
necklace $\mathcal N$ defining the positroid. We denote such set-systems as
$\mathcal{I}nt(\mathcal N )$. In this paper we give an alternative (and
shorter) proof of the purity of $\mathcal{I}nt(\mathcal N )$ and present a
stronger result. More precisely, we introduce a set-system
$\mathcal{O}ut(\mathcal N )$ complementary to $\mathcal{I}nt(\mathcal N )$, in
a sense, and establish its purity. Moreover, we prove (Theorem~3) that these
two set-systems are weakly separated from each other. As a consequence of
Theorem~3, we obtain the purity of set-systems related to pairs of weakly
separated necklaces (Proposition 4 and Corollaries 1 and 2). Finally, we raise
a conjecture on the purity of both the interior and exterior of a generalized
necklace.
