关于数量代数的公理化
On the Axiomatizability of Quantitative Algebras
摘要：
定量代数（QA）是度量空间上的代数，由定量方程理论定义，由同一作者在LICS 2016上发表的相关论文中介绍。这些代数为概率、随机和其他定量系统的度量语义提供了数学基础。本文研究了QA的公理化问题。我们调查的整个频谱类型的定量方程，可用于公理化理论：（一）简单的定量方程;（二）霍恩条款不超过$c$方程之间的变量作为假设，其中$c$是一个基数和（iii）最一般的情况下霍恩条款。在每种情况下，我们的特点类的QA和证明品种/拟品种定理，扩展和推广经典的结果，从模型理论的代数和一阶结构。
英文摘要：
Quantitative algebras (QAs) are algebras over metric spaces defined by quantitative equational theories as introduced by the same authors in a related paper presented at LICS 2016. These algebras provide the mathematical foundation for metric semantics of probabilistic, stochastic and other quantitative systems. This paper considers the issue of axiomatizability of QAs. We investigate the entire spectrum of types of quantitative equations that can be used to axiomatize theories: (i) simple quantitative equations; (ii) Horn clauses with no more than $c$ equations between variables as hypotheses, where $c$ is a cardinal and (iii) the most general case of Horn clauses. In each case we characterize the class of QAs and prove variety/quasivariety theorems that extend and generalize classical results from model theory for algebras and first-order structures.
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。
