During the early days of relational database theory it was realized that "acyclic" database schemas possess a number of desirable properties. In fact, three different notions of "acyclicity" were identified and investigated during the 1980s, namely, α-acyclicity, β-acyclicity, and γ-acyclicity. Much more recently, the study of α-acyclicity was extended to annotated relations, where the annotations are values from some positive commutative monoid. The recent results about α-acyclic schemas and annotated relations give rise to results about β-acyclic schemas and annotated relations, since a schema is β-acyclic if and only if every sub-schema of it is α-acyclic. Here, we study γ-acyclic schemas and annotated relations. Our main finding is that the characterization of γ-acyclic schemas in terms of monotone sequential join expression extends to annotated relations, provided the annotations come from a positive commutative monoid that has the inner consistency property. Furthermore, the results reported here shed light on the role of the join of two standard relations. Specifically, our results reveal that the only relevant property of the join of two standard relations is that it is a witness to the consistency of the two relations, provided that these two relations are consistent. For the more abstract setting of annotated relations, this property of the standard join is captured by the notion of a consistency witness function, a notion which we systematically utilize in this work.

We study the first-order axiomatisability of finite semiring interpretations or, equivalently, the question whether elementary equivalence and isomorphism coincide for valuations of atomic facts over a finite universe into a commutative semiring. Contrary to the classical case of Boolean semantics, where every finite structure is axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and depends on the underlying semiring. We prove that for a number of important semirings, including min-max semirings, and the semirings of positive Boolean expressions, there exist finite semiring interpretations that are elementarily equivalent but not isomorphic. The same is true for the polynomial semirings that are universal for the classes of absorptive, idempotent, and fully idempotent semirings, respectively. On the other side, we prove that for other, practically relevant, semirings such as the Viterby semiring 𝕍, the tropical semiring 𝕋, the natural semiring ℕ and the universal polynomial semiring ℕ[X], all finite semiring interpretations are first-order axiomatisable, and thus elementary equivalence implies isomorphism.