Consider $X$ a finite set and let $2^{X}-\emptyset$ denotes its power set (excluding the empty set).
Definition 1: A choice function is a function $c:2^{X}-\emptyset\mapsto X$ satisfying $c(A)\in A$ for every non empty $A$
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Definition 2: A choice function satisfies $\alpha$ if the following property is satisfied: $$x=c(A)\quad\text{and}\quad x\in B\subseteq A \implies x=c(B)$$ for all non-empty $A, B\subseteq X$
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Definition 3: A choice function satisfies $\gamma$ if the following property is satisfied: $$x=c(A)=c(B)\implies x=c(A\cup B)$$ for all non-empty $A, B\subseteq X$
Consider $c_1,\ldots, c_n$ choice functions satisfying the functional property $\alpha$ defined above.
Let $C:2^{X}\mapsto 2^{X}$ defined as
$$C(A)=\bigcup_{i=1}^{n}c_i(A)$$
We have the following result:
$C$ satisfies the following functional properties
(i) [Chernoff]: $A\subseteq B \implies C(B)\cap A\subseteq C(A) $ $\quad$ $\;\forall\;A, B\in 2^{X}-\emptyset$
(ii) [Aizerman]: $C(B)\subseteq A\subseteq B \implies C(A)\subseteq C(B)$ $\quad$ $\;\forall\;A, B\in 2^{X}-\emptyset$
Moreover (i) and (ii) characterize $C$; in the sense that if $S$ satisfies (i) and (ii) then it is a union of $n$ choice functions satisfying $\alpha$
To see that $C$ satisfies $(i)$ and $(ii)$ it suffices to show that this functional properties are preserved under unions and are satisfied by a choice function with property $\alpha$.
The complete proof of this result can be found in the paper (Theorem 5): H. Moulin. Choice Functions Over a Finite Set: A Summary. Social Choice and Welfare
My question is the following:
What functional property (or properties) are satisfied by $C$ if instead $c_1, \ldots, c_n$ satisfy $\gamma$ (and not $\alpha$) ?
My attempt was to look for a functional property that is satisfied by a choice function with property $\gamma$ and is preserved under unions; but so far any relation that comes to my mind is not satisfied by this new $C$. Any help will be highly appreciated.