AI 中文总结
该研究系统发展线性序的加法算术，通过欧几里得算法解决塔尔斯基1956年提出的线性序加法交换问题，刻画加法交换对及可在有序和中表示的交换半群。
AI 中文摘要
我们系统发展了线性序类在有序和（LO, +）下的算术，并证明了若干新结果。我们的方法基于针对几乎加法交换的线性序对的欧几里得算法。我们的结果包括：（i）对（LO, +）的主要经典定理进行了推广并给出统一证明，包括（LO, +）的林登鲍姆分解定理以及阿隆萨津提出的加法交换线性序对的表示定理；（ii）解决了塔尔斯基在1956年提出的问题：对于每对线性序A、B以及每组四个自然数n, m, k, l ≥ 1，若nA + mB ≅ kB + lA，则必有A + B ≅ B + A，这一命题是否成立？塔尔斯基与张（Chang）已证明该命题对某些系数选择成立，我们证明其在一般情况下成立；（iii）证明了LO中加法交换对的如下刻画：A + B ≅ B + A当且仅当ωA初始嵌入ωB且ω*A最终嵌入ω*B，或反之，我们表明这可视为塔尔斯基一项被证伪猜想的正确修订版；（iv）刻画了可在（LO, +）中表示的交换半群（S, ⊕），并特别证明它们均为克利福德意义下的自然全序交换半群。
英文摘要
We present a systematic development of the arithmetic of the class of linear orders under the ordered sum $(LO, +)$ and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute. Among our results: (i.) We generalize and give unified proofs of the main classical theorems for $(LO, +)$, including Lindenbaum's division theorem for $(LO, +)$ and a representation theorem for additively commuting pairs of linear orders due to Aronszajn. (ii.) We solve the following problem, posed by Tarski in 1956: is it true that for every pair of linear orders $A, B$ and quadruple of natural numbers $n, m, k, l \geq 1$, if $nA + mB \cong kB + lA$ then $A + B \cong B + A$? Tarski and Chang showed the answer is yes for certain choices of the coefficients $n, m, k, l$. We show the answer is yes in general. (iii.) We prove the following characterization of the additively commuting pairs in $LO$: $A + B \cong B + A$ if and only if $ωA$ embeds initially in $ωB$ and $ω^* A$ embeds finally in $ω^* B$, or vice versa. We show this can be viewed as a correctly revised version of a refuted conjecture of Tarski. (iv.) We characterize the commutative semigroups $(S, \oplus)$ that can be represented in $(LO, +)$ and show in particular they are all subsemigroups of naturally totally ordered semigroups in the sense of Clifford.
Comments108 pages
