Motivation. Having read about infinite time Turing machines and ω-languages, I was thinking about more general notions of languages and “computation time”. Languages over strings of length greater than ω seem reasonably easy to define, and using larger ordinals to measure time is quite standard.

However, I’m interested in more general, non necessarily well-founded ordered sets. Using ordered sets to keep track of time seems to require suitable notions of arithmetic operations. For example, it’s easy to generalise sum and multiplication of ordinals to arbitrary totally ordered sets (though, of course, these operations are not defined by transfinite recursion); I’m not sure about exponentiation and other operations.

Question. Is there any good literature about arithmetic of (some classes of) non well-founded ordered sets?