Dedekind cuts of partial orderings

Dedekind cuts are a clever trick for defining the reals given the rationals. Such a cut considers a set C of rationals such that if x is in C and y < x then y is in C. For any such cut there is just one real “between” C and its complement. Further there is just one cut for each real, (unless, of course the real is also rational, then there are two, one including the real and the other not). The only logical property of the rationals used here is their partial ordering. That this partial ordering is also a total (simple) ordering is interesting but not just now. Also that the ordering of the rationals is dense is interesting, but not here.

We may take any partial ordering and consider such cuts. The result is always a lattice.

There is another different unique (within isomorphism) lattice associated with any partial ordering. There is for any partial ordering some unique smallest lattice in which it is embedded. The lattice may contain new elements but the new ordering, restricted to the old PO will contain no new orderings. This construction is also found in security considerations. The orange book provides a theory of security classifications that implicitly defines a lattice. In a particular computer system it is likely that some of the lattice values will be unused. This may cause some confusion. It should not any more than noting that the boolean or command of the CPU need not in an application produce all possible values in order to make the set of all possible values a useful concept with which to reason. It is the same with the lattice of security classifications.

When the partial ordering is finite and total the cuts add nothing of interest.