There are several ways to combine different preference criteria--or, in
other words, building new posets from existing posets. The Noos operations
dealing with preference combination are methods that create new posets
from (a combination of) posets--posets created by preference methods.
We will show here the
*hierarchical union* of posets, that is tantamount to
consider some preferences more important (more preferred) than others. Let us
imagine we prefer PCs with PowerPC chip *and* we prefer cheaper PCs.
If we take both preference criteria as equal, we can construct the poset
that is the *union* of both posets.
Given two preference defined over the
same set *A* we define their *transitive union preference* as

where is the transitive closure of .

The result in our example is that we prefer both
`PC-white` `PC-red` (the cheaper) and `PC-red` `
PC-white` (the PowerPC is preferred). This is equivalent to build a poset
as the *poset union* of the results of `cheaper-pref` and `
ppc-pref`. The problem now having a cycle in preferences is that we are
incapable of deciding (preferring) `PC-white` or `PC-red`. The
hierarchical union `h-union` of preferences can be considered as a preference
over preferences such that, after some order among pairs of elements
have been is added by a preference, the next preference method is
allowed to add new order only among pairs that are not already ordered.
Formally, given two posets defined over the
same set *A* the *hierarchical union* poset is defined as follows:

Forbidding the addition of when already has been
set by a prior (more preferred) preference simply we avoid cycles. Indeed, we can
consider hierarchical union of preferences as a process of refining a
partial order by means of new preference criteria. This is shown in the next
example where the `h-union` method embodies the hierarchical union of
`higher-poset` and `lower-poset`.

The `subsumption-preference` takes the unordered set of PCs and adds
the preferences for PowePC computers, and returns the appropriate poset
where `PC-blue` and `PC-white` are preferred to `PC-red`. Now
this poset is the value of `higher-poset` feature of `
h-union` that adds the cheaper preferences of `lower-poset` only when a
higher preference is not already established. In this situation, only a
preference among `PC-blue` and `PC-white` can be added, and since
`PC-blue` is cheaper it is preferred to `PC-white`. The final
result of `ppc-cheaper-pref` is then poset
{`PC-blue` `PC-white` `PC-red`} that
contains no cyclic preferences.

Thu Jan 23 11:36:28 MET 1997