Definition. Compositionality, computability, and complexity › Semantic systems (§ 3) [pagin2021-semantic-system]

Given a grammatical term algebra $GT{A}_L$ and domain $M$ of meanings, a semantic system $S$ is a triple $⟨S^{\prime },{\mu }_S,{\mathrm{\Phi }}_S⟩$ where—

1. $S^{\prime }$ is a finite set of semantic functions ${\mu }_j:G{T}_L\to M$, ${\mu }_S\in S$, and
2. ${\mathrm{\Phi }}_S:S^{\prime }\times {\mathrm{\Sigma }}_L\times N\to S^{\prime }$ is a partial selection function.

Given some semantic function ${\mu }_j$ and a syntactic operation $\sigma$, ${\mathrm{\Phi }}_S$ then gives another semantic function ${\mu }_k$ for each $n$th-position term.

Finally, $S(t)={\mu }_S(t)$.