Yes, in a paper a few years back (see
http://groucho.gsfc.nasa.gov/Code_520/Code_522/Tech_Collab/univ_contacts/
joslyn/papers.html#JoC91c) I formally defined a hierarchy as any partially
ordered structure. These admit to descriptions in terms of levels, while
not being so strict as to require that each element have only a single
parent. The latter are also hierarchies, but I call them "strict
hierarchies". Graphically, the two classes of hierarchy correspond to
directed acyclic graphs and trees respectively.
As you say, the essential aspect is the absence of cycles. Whenever a cycle
is present, the nodes which participate in the cycle are amalgamated into a
single new meta-node, and the (loose) hierarchical structure is recovered.
If the cycle goes from a leaf to the root, then clearly under this
procedure the whole structure collapses into a single meta-node.
>I think the only way to capture the concept of "heterarchy" is to consider
>the simultaneous interwovenness of different views of the operation of a
>system.
I regret that McCulloch's heterarchy paper has been on my "to read" stack
for far too long. My impression from talking to others has been that he
does not give a formal description for a heterarchy, for example as any
directed graph, cyclic or not, but rather describes it in terms
specifically of a control heterarchy, where the "locus of control" is free
to move among the nodes. But please correct me.
>The concept of hierarchy has been formally defined in the following
>article, which may be useful:
>
>R.Ramal, G.Toulouse and M.A.Virasoro (1986)
>"Ultrametricity for Physicists", Review of Modern Physics 58,
>pages 765-788.
>
>"Ultrametricity" is a property of hierarchies which is similar to the
>law of transitivitiy in logic.
Interesting. Is this a formal definition, or one tied to physical systems
(being in a physics journal)?
O---------------------------------------------------------------------------->
| Cliff Joslyn, NRC Research Associate, Cybernetician at Large
| Mail Code 522.3, NASA Goddard Space Flight Center, Greenbelt, MD 20771 USA
| joslyn@kong.gsfc.nasa.gov http://pespmc1.vub.ac.be/joslyn 301-286-5773
V All the world is biscuit-shaped. . .