The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The result does not extend directly to polynomials over semirings, but we do have analogous results for some special semirings, for example, the tropical, extended and supertropical semirings. These all fall into a larger class of upper-bound semirings. In this paper we extend the known results and give a complete characterization of elementary upper-bound semirings. We further improve this characterization statement in the case of linearly ordered upper-bound semirings.

环上对称多项式的基本定理是一个经典结果，它表明每个单位交换环都是基本的，即我们可以用独特的方式用基本的代数表达对称多项式。结果并未直接扩展到半环上的多项式，但是对于某些特殊的半环，例如热带，扩展和超热带半环，我们的确具有类似的结果。所有这些都属于上边界半环的一大类。在本文中，我们扩展了已知的结果，并给出了基本上限半环的完整表征。在线性排序的上半环的情况下，我们进一步改进了该表征语句。