# Are all subcorrespondences of the weak Pareto correspondence monotonic at the unrestricted domain of linear orders?

I have a doubt regarding the well-known concepts of weak Pareto optimality and monotonicity.

Let $$N$$ be a finite set of players, let $$A$$ be a finite set of alternatives, let $$\mathcal{P}$$ be the set of all linear order profiles on $$A$$, and let $$F:\mathcal{P}\to 2^A\backslash\{\emptyset\}$$ be a social choice correspondence.

Let $$F^*:\mathcal{P}\to 2^A\backslash\{\emptyset\}$$ be the weak Pareto correspondence: namely, for all linear order profiles $$P\in\mathcal{P}$$, $$\begin{gather} F^*(P)=\{x\in A\mid(\nexists y\in A)[(\forall i\in N)(yP_ix)]\} \end{gather}$$

Given any player $$i\in N$$, any alternative $$x\in A$$ and any linear order profile $$P\in\mathcal{P}$$, let $$L_i(x,P)=\{y\in A\mid xP_iy\}$$ be player $$i$$'s lower contour set at $$x$$.

A social choice rule $$F:\mathcal{P}^N\to 2^A\backslash\{\emptyset\}$$ is monotonic if and only if for all alternatives $$x\in A$$ and all linear order profiles $$P,P'\in\mathcal{P}$$, the following is true: if $$x\in F(P)$$ and $$L_i(x,P)\subseteq L_i(x,P')$$ for all $$i\in N$$, then $$x\in F(P')$$.

We know by Maskin & Sjöström (Footnote 15, p. 248, 2002) that the weak Pareto correspondence is monotonic at the unrestricted domain of linear orders.

What I am wondering is whether all subcorrespondences of the weak Pareto correspondence is also monotonic at the unrestricted domain of linear orders.

Let $$A=\{a,b,c\}$$. Consider the sub-correspondence of the weak Pareto correspondence in which $$c$$ is removed unless $$c$$ is the only weak Pareto optimum or $$b$$ is a weak Pareto optimum. There are two agents. The value under the profile consisting of $$a\succ b\succ c$$ and $$c\succ b\succ a$$ is $$\{a,b,c\}$$. However, under the profile consisting of $$a\succ c\succ b$$ and $$c\succ b\succ a$$, the value is $$\{a\}$$. So this sub-correspondence is not monotonic.