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 can obviously be axiomatised up to isomorphism by a first-order sentence, the situation in semiring semantics is rather different, and strongly 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, the tropical semiring, the natural semiring and the universal polynomial semiring N[X], all finite semiring interpretations are first-order axiomatisable (and thus elementary equivalence implies isomorphism), although some of the axiomatisations that we exhibit use an infinite set of axioms.

翻译：我们研究的是有限半环解释的第一阶惯性,或者同等的,基本等值和形态论是否与对有限宇宙的原子事实的估价一致,形成一种通融半分。 与布利安语义学的古典案例相反,在布利安语语系中,每个有限结构显然都可以通过一阶的句子解解为异形,半半环语系的情况相当不同,并在很大程度上取决于基本半分系。我们证明,对于一些重要的半分系,包括微轴半分环和正面布林表达的半分系,存在着一些基本等同但非非非无异形的限定半分系解释。对于分别适用于吸收、极能和完全同系半分系的多种混合结构来说,同样的情况是。 另一方面,我们证明,对于其他重要的半分系、热带半环系、自然半环系和正布林系半环系等重要半分系的半分系,一个普遍的多环系类半成(我们可理解的初等分系的半系)的半成像(我们可理解的半氧化的半氧化的半成一等定的半成一面的半成一面的半成一面)等。