The structure of k-CS-transitive cycle-free partial orders with infinite chains

Abstract

The notion of cycle-free partial order (CFPO) was defined in [9] and the class of sufficiently transitive CFPOs was investigated, and in many cases, classified. In [10] a complete classification was given for the countable 3- or 4-CS-transitive CFPOs having a finite chain and embedding an infinite ‘alternating chain’ ALT. Here, the major case is that of the so-called ‘skeletal’ CFPOs, in which the CFPO itself forms a mere skeleton of the structure of the whole picture, which is more accurately provided by its Dedekind–MacNeille completion. It was asserted that a similar classification should also be possible in the infinite chain case (still with the countability restriction), and it is the object of the present paper to finish this task.

The overall plan of the work is similar in style to [10]. The CFPOs we study can all be construed as arising from a chain, suitable ‘adorned’ with instructions as to how to branch (and repeatedly). An obvious difference from the finite chain case is that, this time, many of the points in the chain will actually survive as elements of the structure; whereas in [10], since the chains of the structure had to have length 2, only the endpoints would. On the whole, the treatment is parallel, with just a small number of additional cases.

We recall the main ideas from [10], quote results that are needed, and concentrate on aspects of the structures which are new or specific to the infinite chain case. We conclude by remarking on some direct connections between the finite and infinite chain cases.