CYCLE-FREE PARTIAL ORDERS 3

Fig. 1.2.

With n = 4 the a,- shown constitute a cycle, and so since non-adjacent a,s

are incomparable, this is not a CFPO according to the definition. A limited

class of non-trees does satisfy the definition; for instance the partial order

shown in figure 1.3. In fact a CFPO in our sense (given below) fulfils the

above definition if and only if there are no four points ao, a\y a^, as related

in precisely the same way as in figure 1.2.

Instead of living with this restriction, we prefer to modify the definition

so as to capture the class which it appears was intended in the first place.

Partly this is because it seems likely that it is in some sense a maximal class

to which Rubin's reconstruction methods for trees may apply. Specifically

it will allow us sufficient freedom to carry out 'piecewise patching', meaning

that we may glue together isomorphisms between (convex) pieces so that

their union will be an automorphism.

The definition of CFPO is given in terms of a suitable completion,

called the 'Dedekind-MacNeille completion' (or sometimes just 'Dedekind-

completion' for short) of the original partial order. As motivation for taking

Fig. 1.3.