This is Part 2 of a guest post by Sam Corson, who is a Heibronn Fellow at the University of Bristol. It will be helpful to read Part 1 first.
We will furthermore overload the notation used for word concatenation and apply it to totally ordered sets. For example, we will write for totally ordered sets and provided there exists an order isomorphism between them. The concatenation of totally ordered sets and is denoted and is the disjoint union under the obvious order. If is a collection of totally ordered sets, indexed by a totally ordered set , then the concatenation is the totally ordered set which is the disjoint union under the natural order.
Given a word and there exists a maximal interval such that and is -pure. We can therefore write where each is a maximal nonempty interval in for which is -pure, and the totally ordered set is unique up to . This gives rise to a decomposition of the word as where . This decomposition we call the p-decomposition and write to express that the p-decomposition of is the concatenation . We let denote the totally ordered set , which is well-defined up to . Of course, . As an example, one can consider the word
where one has and where .
A word is a p-chunk of the word provided there exists some interval such that (we may indeed write ). Thus a p-chunk of a word is a subword which respects the p-decomposition. Given an interval we write for the p-chunk . An -pure p-chunk of a word will clearly either be or will be one of the . Given we will let denote the set of all p-chunks of . Note that this set might be uncountable (if then consider the p-chunks associated with the Dedekind cuts). Given a subset we let denote the generated subgroup . One can prove that this subgroup is closed under taking p-chunks of elements.
For there similarly exists a decomposition of into maximal nonempty intervals where is -pure. Thus we obtain a decomposition, which we again call the p-decomposition and use the same notation and again write to identify the p-index.
If with and is finite then we have , and similarly for a word . This fact that the class is preserved under deleting finitely many elements of the p-index and then reducing provides the motivation for the essential idea in constructing the isomorphism. We recount the idea now.
Given a totally ordered set we will say that a subset is close in if for every infinite interval we have . For example, if is finite then every subset of , including , is close in . If then any infinite subset is close in . If then every dense subset of will be close in . If and are totally ordered sets and and are each close then we call an order isomorphism a close order isomorphism (abbreviated coi) from to .
A close order isomorphism from to defines a correspondence (not necessarily one-to-one) between the intervals of and those in : given interval we obtain interval . For an interval we define similarly. Many nice properties hold for this correspondence; for instance, we have is a subinterval of and there exist (possibly empty) finite subintervals such that .
If and and is a coi from to then we write and call such a triple a coi triple. A collection of coi triples is coherent if for any choice of , intervals and , and such that
we get that
and also for any choice of , intervals and , and such that
we get that
Note that it is possible that a collection of coi triples which has only one element can fail to be coherent, since the above definition allows that and . Thus, great care must be taken in producing a coherent collection of coi triples. For a coherent example, one can take
where we clearly have and let be the unique order isomorphism between and . Of course, one needs to check that the appropriate conditions hold in order to conclude that is coherent. As a hint in this easy example, one can see that if are intervals and and either of or is finite, then both are finite and . If at least one of or is infinite then both must be infinite and in fact implies and the desired equality once again holds. One cannot have if either of or is infinite (by considering the order type), and if either of or is finite we again see that . The check for intervals in is comparable.
One can imagine that the check for coherence becomes annoying when the collection has more elements and words become more complicated. The payoff for producing such a collection, however, is hinted at in the following:
Proposition. Let and denote the respective quotient maps. A coherent collection of coi triples induces an isomorphism
If we can produce a coherent collection of coi triples which is plentiful enough that
then we have obtained an isomorphism.
Although the proposition is very intuitive, the proof is technical. One first shows that from a coherent collection one obtains well-defined homomorphisms and . That these natural maps (extending ) are well-defined requires some effort, since a word might decompose in many distinct ways as a finitary concatenation of elements in , and coherence is essential to the argument. Once these homomorphisms are in hand, it is easier to see that they descend to homomorphisms and and that these homomorphisms are inverse to each other.