TITLE: Sets of integers in which every element is the sum of two others
QUESTION [10 upvotes]: i) Does there exists a nonempty subset of integers $S$, such that for all $a \in S$, there exists $b,c \in S$ such that $a = b + c$, where $a, b, c$ are distinct integers. 
Edited to add:
ii) Can an $S$ with these properties be finite, and also have the property that $a \in S$ implies $-a \notin S$
This question is inspired by another question, which has not been fully answered yet.

REPLY [2 votes]: A further answer, generalizing flagar's example: 
For any N ≥ 10, there exists a set of the sort you ask for which contains at most eleven elements (with fewer than eleven in the case that N < 13), and with all elements having absolute value at most N:

{−N, −N+2, −N+4, −2, 1, 3, 4, 5, N−7, N−6, N−5}.

The shorter example presented by flagar corresponds to the case N = 10.
The key to this construction was to think in terms of small sequences of almost mutually-supporting subsequences. For instance, the sequence −N, −N+2, −N+4 consist of integers which are sums of themselves, plus one of −2 or 4. We can include 4 in the set by including it in the sequence 3, 4, 5: we can support this by including 1 in the set (where 1 itself is supported as the sum of 3 and −2). We then only need a way of supporting −2: we do this with the other subsequence which includes N−6.
Each time we have a short sequence, we have a collection of integers which is easy to "support" using only a small number of other integers. So one strategy to constructing such self-supporting sets of integers is to think in terms of one or more short sequences of integers, with each sequence working to support another.