A lot has been written about the arithmetic of ordinal numbers. However, we can **also** do arithmetic with linearly ordered sets.

Question.Is there an article or book where I can learn the basics of linearly ordered set arithmetic?

Here's the definitions I have in mind. Let $I$ denote a totally ordered index set, and suppose $L$ is an $I$-indexed family of linearly ordered sets. Then:

**Definition 0.** The summation $\sum_{i \in I} A_i$ is defined in the same way as for ordinals; in particular, it is the linearly ordered set consisting of all the $A_i$'s stuck head-to-tail in the order they're indexed. Further, we define $A \times B$ as shorthand for $\sum_{b \in B} A$ whenever $A$ and $B$ are totally ordered sets.

**Definition 1.** The product $\prod_{i \in I}A_i$ is defined **only if** $I$ is in fact well-ordered, in which case it is the linearly ordered set whose underlying set is the obvious product of sets, and whose order relation is defined lexicographically. Further, we define $A^\beta = \prod_{\alpha \in \beta} A$ whenever $A$ is a linearly-ordered set and $\beta$ is well-ordered.