哪些半域是正合的？
Which semifields are exact?
摘要：
域上有限维向量空间的子空间上的每一个（左）线性函数都可以延拓为整个空间上的一个（左）线性函数。本文探讨了线性代数的这一基本事实适用于更一般结构的程度。在线性函数上具有类似性质的半域被称为（左）精确的，我们提出了这样的半域的完整描述。也就是说，我们证明了半域$S$是左正合的当且仅当$S$是除环或幂等半环。特别是，我们的结果是新的，甚至热带半环，并给出了一个解决问题所提出的Wilding。并指出了需要进一步研究的几个问题。
英文摘要：
Every (left) linear function on a subspace of a finite-dimensional vector space over a (skew) field can be extended to a (left) linear function on the whole space. This paper explores the extent to what this basic fact of linear algebra is applicable to more general structures. Semifields with a similar property imposed on linear functions are called (left) exact, and we present a complete description of such semifields. Namely, we show that a semifield $S$ is left exact if and only if $S$ is either a skew field or an idempotent semiring. In particular, our result is new even for the tropical semiring and gives a solution to the problem posed by Wilding. Also, we point out several problems that require further investigation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
