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Source. “The Generating Theorem,” PDF pp. 1--2 of the author survey. The survey identifies it as Theorem 1 of Zhi-Wei Sun, Systems of congruences with multipliers, Nanjing Univ. J. Math. Biquarterly 6 (1989), no. 1, 124--133.

Conventions

Let $M$ be an additive commutative monoid, let $\Lambda\subseteq M$, and let

$$A=\{(\lambda_s,a_s,n_s)\}_{s=1}^k, \qquad 0\leq a_s<n_s,$$

be a finite system in which $\lambda_s\in\Lambda$ weights the residue class $a_s+n_s\mathbb Z$. Its covering map is

$$w_A(x)=\sum_{s=1}^k\lambda_s\chi_s(x),$$

where $\chi_s(x)=1$ when $x\in a_s+n_s\mathbb Z$ and is $0$ otherwise. The sum-system $A\sqcup B$ retains all triples, including repetitions. For $S\subseteq M$, let $S_*$ be the class of systems $A$ with $w_A(\mathbb Z)\subseteq S$, and write

$$R(q)=\{0,1,\ldots,q-1\}.$$

Statement

All systems in $S_*$ are generated by the following two operations.

1. If $\lambda_1,\ldots,\lambda_k\in\Lambda$ and $\lambda_1+\cdots+\lambda_k\in S$, then

$$\{(\lambda_s,0,1)\}_{s=1}^k\in S_*.$$

2. Let $q$ be prime. For every $r\in R(q)$, suppose

$$A_r= \{(\lambda_s,a_{sr},n_s)\}_{s=1}^k \sqcup \{(\lambda_j^{(r)},a_j^{(r)},n_j^{(r)})\}_{j=1}^{h(r)} \in S_*,$$

with $\max_{r\in R(q)}h(r)>0$. Suppose also that every $n_s$ is coprime to $q$ and, for each $s$, there is a unique $a_s\in R(n_s)$ satisfying

$$a_s\equiv r+q a_{sr}\pmod{n_s} \qquad\text{for every }r\in R(q).$$

Then

$$A=\{(\lambda_s,a_s,n_s)\}_{s=1}^k \sqcup \bigsqcup_{r=0}^{q-1} \{(\lambda_j^{(r)},r+q a_j^{(r)},q n_j^{(r)})\}_{j=1}^{h(r)}$$

belongs to $S_*$.

Proof scope. This is the survey's statement-level restatement of an earlier theorem. The 1989 primary paper and its proof were not acquired or independently checked here.