a homomorphism ${\mathbf{A}}\rightarrow {\mathbf{B}}$.

The converse of the condition given in this definition always holds: suppose that $h$ is a homomorphism from ${\mathbf{A}}$ to ${\mathbf{B}}$. Then, the instance $({\mathbf{A}}, {\mathbf{B}})$ has the SACC via the mapping $s$ where $s(a) = \{ h(a) \}$ for all $a \in A$ and the mappings $h_{a, b}$ defined by $h_{a, b}(a') = \{ h(a') \}$ for all $a' \in A$.

We use the notation ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^n)$ to denote the induced substructure of ${\wp}({\mathbf{B}})^n$ whose universe contains an $n$-tuple $(S_1, \ldots, S_n)$ of ${\wp}({\mathbf{B}})^n$ if and only if it holds that $\bigcup_{i \in [n]} S_i = \bigcup_{i \in [n], |S_i| = 1} S_i$.

\[thm:sac-solves\] Let ${\mathbf{B}}$ be a structure. Singleton arc consistency solves ${{\mathsf{CSP}}}({\mathbf{B}})$ if and only if for all $n \geq 1$ there is a homomorphism ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^n) \rightarrow {\mathbf{B}}$.

First we show that if singleton arc consistency solves ${{\mathsf{CSP}}}({\mathbf{B}})$, then there is a homomorphism from ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$ to ${\mathbf{B}}$ for all $n\geq 1$. Let $n \geq 1$; we show that $({\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n}),{\mathbf{B}})$ has the SACC. Then, there is a homomorphism from ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$ to ${\mathbf{B}}$, since the singleton arc consistency algorithm solves [CSP(${\mathbf{B}}$)]{}.

Let $s$ be the mapping $s(a):=\bigcup_{i \in [n]} S_i$ for all tuples $a=(S_1,\dots,S_n)$ of ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$. 