标题： 单位系统的数学结构 显示英文标题

标题： Mathematical structure of unit systems

摘要： 我们研究单位系统的数学结构及其之间的关系。 纵观所有单位系统，我们可以找到一种称为预序（或准序）的数学结构。 对于某些单位系统对，存在一种预序关系，使得一个单位系统可以转换为另一个单位系统。 只有当后一个系统中可区分的所有量在前一个系统中始终可区分时，这种转换（或转换）才是可能的。 利用这种结构，我们可以系统地比较不同单位系统中的表示。 特别是，单位系统的等价类（EUS）起着重要作用，因为在属于同一EUS的单位系统中，物理量和方程的表示形式是相同的。 每个EUS中量的维度都是唯一定义的。 EUS形成一个偏序集。 利用这些数学结构，单位系统和EUS被系统地分类和组织成一个分层树。 显示英文摘要

摘要： We investigate the mathematical structure of unit systems and the relations between them. Looking over the entire set of unit systems, we can find a mathematical structure that is called preorder (or quasi-order). For some pair of unit systems, there exists a relation of preorder such that one unit system is transferable to the other unit system. The transfer (or conversion) is possible only when all of the quantities distinguishable in the latter system are always distinguishable in the former system. By utilizing this structure, we can systematically compare the representations in different unit systems. Especially, the equivalence class of unit systems (EUS) plays an important role because the representations of physical quantities and equations are of the same form in unit systems belonging to an EUS. The dimension of quantities is uniquely defined in each EUS. The EUS's form a partially ordered set. Using these mathematical structures, unit systems and EUS's are systematically classified and organized as a hierarchical tree.