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.