Let $\mathcal P$ be the set of finite subsets of $\mathbb Z_{\geq 0}$ that contains $0$. We say that $A \in \mathcal P$ is indecomposable if it is not $B+C$ with $B,C\in \mathcal P$ and $B,C\neq \{0\}$.

It is easy to see which small cardinality sets are indecomposable. If $|A|=2$, then it is indecomposable. If $A=\{0<x<y\}$, A is indecomposable iff $y\neq 2x$. If $A=\{0<x<y<z\}$, A is indecomposable iff $z\neq x+y$.

My questions are:

Questions: have these sets been studied before? Are there nice characterizations? Do they have interesting counts, for instance, the number of indecomposable sets whose largest element is $n$?

Let $\mathcal P$ be the set of finite subsets of $\mathbb Z_{\geq 0}$ , each of them contains $0$. We say that $A \in \mathcal P$ is indecomposable if it is not $B+C$ (the sum set of $B,C$) with $B,C\in \mathcal P$ and $B,C\neq \{0\}$.

It is easy to see which small cardinality sets are indecomposable. If $|A|=2$, then it is indecomposable. If $A=\{0, x, y\}$ with $0<x<y$, A is indecomposable iff $y\neq 2x$. If $A=\{0<x<y<z\}$, A is indecomposable iff $z\neq x+y$.

My questions are:

Questions: have these sets been studied before? Are there nice characterizations? Do they have interesting counts, for instance, the number of indecomposable sets whose largest element is $n$?