6

EXTENDED INTRODUCTION

we also write [a, b] for {x : b x a] when 6 a. A connecting set from a

to b in M is a finite sequence (ao, a i , . . . , an) such that a = ao, 6 = an, and

for each r n, ar and ar+i are comparable (either ar ar+i or ar+i a

r

).

A path from a to b is a set of the form Ao U Ai U At U ... U A

n

_i where

for some connecting set ao, ai, 02,..., an from a to 6, each Ar is a maximal

chain in [a

r

,a

r

+i], and for distinct r,s, the only non-empty intersections

between lr and A$ are given by ^4r H -Ar+i = {ar+i}«

After these preliminaries we can now define what it means for a Dedekind-

complete partial order to be cycle-free. We say that (M, ) is a cycle-free

partial order if for every a and b in M there is a unique path in M from

a to 6. More generally, for any partial order (M, ), we say that it is a

cycle-free partial order if its Dedekind-completion (M

D

, ) is.

The fact that we have to pass to the completion of M in order to tell

whether or not it is a CFPO may seem rather unsatisfactory, but in our

view this gives the most workable and coherent definition, and from the

rather striking examples of CFPOs presented in this paper, most of which

are described very much in terms of the relationship between M and

MD,

the approach seems justified.

In [1] some discussion is given of 'acyclic' or 'cycle-free' partial orders.

An alternative definition is proposed, and it is suggested that the following

are desiderata for any such notion:

(1) subsets of acyclic partial orders should be acyclic;

(2) semilinear orders (both upper and lower) should be acyclic;

(3) the diagram representing such an order should be 'tree-like';

(4) the notion should be first-order expressible;

(5) the Dedekind-MacNeille completion of an acyclic partial order should

be acyclic.

Now it will be seen that some at least of these requirements are indeed

met by the notion we have introduced. Namely (5) precisely corresponds

to the way it has been defined. The facts that (2) and (3) apply to our

definition will be amply illustrated throughout the paper. Moreover it is

possible for us to demonstrate the truth of (1) as well, though it is perhaps

not immediately obvious. Undoubtedly it is requirement (4) which causes

the most trouble.

Now from the definition of CFPO given it is clear that any such partial

order must be connected, meaning that there is a path in the partial order

between any two points. It is hardly surprising that this prevents the notion

being first-order expressible. If however we relax this condition, and instead

(just temporarily) say that (M, ) is a: (not necessarily connected) CFPO

if for any two points there is at most one path between them, we obtain the

class of partial orderings which are disjoint unions of connected CFPOs

(and with no comparabilities between points in different components). In

[27] it is shown that this class is first-order axiomatizable, and this is

really the best that one can hope for. Moreover, an axiomatization using