AI 中文总结
研究对称中点代数（尺度）的基本性质，包括其分类，可消去与不可消去的情况及转化，还定义了类似阿贝尔群的张量积，使尺度范畴成为对称幺半闭范畴，并引入‘尺度环’概念。
AI 中文摘要
本文研究了一种称为对称中点代数的代数结构的一些基本性质，该术语‘尺度’借鉴自彼得·弗雷德。尺度大致分为可消去的和不可消去的；前者恰好是二进有理数环上模的子尺度，包括自由尺度；后者通过取商尺度可变为可消去的。此外，可类似阿贝尔群定义尺度的张量积，在此意义下尺度范畴是对称幺半闭范畴。我们将其幺半对象称为‘尺度环’，因为环是阿贝尔群范畴中的幺半对象。
英文摘要
In this paper, we investigate some basic properties of algebraic structures called symmetric midpoint algebras, for which we borrow the term "scales" from Peter Freyd. Scales are broadly divided into cancellative ones and non-cancellative ones; the former are precisely subscales of modules over the ring of dyadic rational numbers, and include free scales; the latter can be turned cancellative by taking quotient scales. Furthermore, we can define tensor products of scales similarly to those of abelian groups, with respect to which the category of scales is a symmetric monoidal closed category. We shall call its monoid objects "scalic rings" after the fact that rings are monoid objects in the category of abelian groups.
