Motivated by enumeration problems, we define linear orders on Cartesian products and on subsets of where each component set is or , ordered in the natural way. We require that be isomorphic to if it is infinite. We want linear orderings of such that, in two consecutive tuples and , at most two components differ, and they differ by at most 1.
We are interested in algorithms that determine the next tuple in by using local information, where “local” is meant with respect to certain graphs associated with . We want these algorithms to work as well for finite and infinite components . We will formalise them by deterministic graph-walking automata and compare their enumeration powers according to the finiteness of their sets of states and the kinds of moves they can perform.
"On defining linear orders by automata." Mosc. J. Comb. Number Theory 9 (3) 253 - 291, 2020. https://doi.org/10.2140/moscow.2020.9.253