Stability under supremum of sets of social choice function with single peaked preferences

Question

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455.

The setting is as follows:

• A non-empty finite set of individuals $N$ indexed by $i=1,\dots,n$.

• The set of alternatives is the real line $\mathbb{R}$.

• Agents are endowed with single-peaked preferences represented by some utility function $u\in S$ over $\mathbb{R}$, where $S$ the set of utility representation of single-peakened preferences is defined as follows:

  $u\in S$ if and only if there exists an alternative $x(u)$, the "peak" of $u$ such that for all $x,y\in \mathbb{R}$

  • $x\leq y < a \Rightarrow u(x)\leq u(y) < u(a)$
  • $a <x \leq y \Rightarrow u(a) > u(x) \geq u(y)$.

• A social choice function $C:\mathbb{R}^n\rightarrow \mathbb{R}$ associates every profile of peakes $(x(u_1),\dots,x(u_n))$ with a chosen alternative.

• A social choice function is said to be strategy-proof if for all $i\in N$, $u_i\Big(c\big(x(u_i),x(u_{-i})\big)\Big) \geq u_i\Big(c\big(x(u_i'),x(u_{-i}\big)\Big)$, for every $u_i' \in S$, where $x(u_{-i})$ is the $(n-1)$-tuple of peaks announced by agents other than $i$.

Here is my question. At the end of the article, Moulin argues that

"$\Sigma_n$, the set of strategy-proof voting schemes [i.e. social choice functions], is stable by the operation of supremum and infimum (not necessarily finite)."

Could anyone help me understanding what stability under supremum and infimum operation actually means (in this context and in general)? I've googled the concept but could not find anything helpful. Two questions in particular:

1. Does "stability of a set under some operation" simply means that the set of is unchanged when the operation is applied to it?

2. How do you apply the supremum and infimum operation to functions? I understand that functions are sets but how does the supremum/infimum operator apply to set which are cartesian products of other sets?

Answer 1

To complement Michael Greinecker's helpful comment (and to be sure that I understand correctly):

1. A property shared by a family of functions is stable under some operation if applying this operation yields a family of functions which also have this property
2. A more detailed definition of pointwise suprema can be found at http://www.proofwiki.org/wiki/Definition:Pointwise_Supremum