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



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



Linearly ordered set arithmetic: reference request
Try
Rosenstein: Linear orderings


<http://books.google.com/books/about/Linear_orderings.html?id=y3YpdW-sbFsC>


