Basic Conceptual Framework of Hierarchies <br>
 A Mathematicial View<br>
 Greg Buzzard and John Erik Forn&#230 ss<br>

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Section 1: Introduction<br>

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There are traditionally three ways of relating to
our world. We have science, philosophy and religion.
It is natural to assume that each of the three approaches
captures some basic aspects of the same reality.

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The purpose of this short note is to provide a conceptual
framework in which these features are naturally intertwined.
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Science is used to describe the <b> Properties</b> of the Universe,
such as how a structure is composed of simpler substructures.
Religion is concerned more with the direction development takes,
so it is teleological, oriented towards <b> Goals</b>.
Philosophy is concerned with analyzing the <b> State</b> of the Universe beyond
what can be done scientifically. Philosophy deals with
the big questions but also with day to day problems.

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In this note we use the concept of a Hierarchical Structure to
unify the three approaches. The Hierarchy consists of Units.
These units have <b> Properties</b> which can be described in a bottom-up
fashion by science and they have <b> Goals</b> which are top-down.
The  Goals and the  Properties come together via the analysis of the
<b> States</b> of the units and how these develop.
 
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Section 2. The parts of a Hierarchical Structure

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A Hierarchy is a finite collection of <b> Units</b> U. There is one
Distuinguished Unit U0. We call U0
the <b>Top Unit.</b>
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Some of the other Units are called <b> Elementary Units</b>. We let
 E denote the collection of Elementary Units. We assume that
 E is not empty and that U0 is not in E <br>

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Each Unit U belongs to a <b> Group </b> G(U) of Units. The Groups are
either disjoint or equal. Hence they partition the Hierarchy into
disjoint Groups. We suppose that G(U0)=U0.

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To each Unit U except U0 there is associated a <b> Superior Unit</b>,
Sup(U), not in  G(U). Moreover, Sup(U)= Sup(V) if and only if
U,V are in the same Group.

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We assume that each Unit U which is not Elementary, is the Superior Unit
of some Group G(V). The members of G(V) are said to be
<b> Subordinate Units</b> to U. We write Sub(U)= G(V).

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Finally we suppose that the Hierarchy is <b> Indecomposable</b> i.e.
that if U is any Unit, then one of the following is the Top Unit:
U, Sup(U), Sup(Sup(U)),... . This final hypothesis is equivalent to
the natural assumption that there are no <b>loops in the chain of command</b>,
 such as U1 being
 the Superior of U2,... ,UN being the Superior of U(N+1) and
 U(N+1) being the Superior of U1.

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Section 3. Properties<br>

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Each Unit U in a Hierarchy is said to have a set of <b> Properties</b>,
P(U). We assume that given any Unit U with Subordinate Units
Sub(U)= (V1, ..., Vn), the Properties P(U) are derived 
from the Properties P(V1),..., P(Vn).

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Section 4. Goals <br>

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Each Unit in a Hierarchy is said to have a set of <b> Goals</b>,
GL(U). We assume that given any Unit U with Subordinate Units
Sub(U)=(V1,..., Vn), the Goals GL(V1),..., GL(Vn)
are derived from the Goals GL(U).

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Section 5. States<br>

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 Each Unit in a Hierarchy has a <b> State</b>,
 St(U).

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The Units V1,..., Vn in a Group Sub(U) 
relate to each other. The States of the Units in a Group
express how they relate. The State of a Unit also
depends on its Subordinates and on its Superior.

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Section 6. General Remarks on Hierarchies

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Let us consider a Hierarchy with Units U. The Units have
Properties P(U), Goals G(U) and States St(U).

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Subsection 6.1 Properties

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One basic remark is that in order to discuss Hierarchies
efficiently, it is useful if the Properties are described
numerically. So we will assume that each P(U) consists
of a finite list of numbers.

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Another basic remark about Properties is that they are
not precisely <b>knowable</b>. There are always errors which
are in the nature of things. Hence, the list of numerical
values for Properties should include an estimate of the
precision of these values, such as the number of accurate
digits.

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If the unit U has Subordinates Sub(U)= V1,..., Vn,
then the Properties P(U) come
from the Properties P(V1),..., P(Vn).
Since the Properties P(Vi) have errors, these errors
transmit to errors in P(U). 

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To have an efficient description
of the Hierarchy, it is useful also to describe the
dependence of P(U) on the properties of its Subordinates
in terms of functions,<br>


P(U)= S(P(V1),...,P(Vn))<br>


In writing down the function S, one usually will make simplifying
assumptions so as to make calculations more feasible. Hence one
introduces <b> Modeling Errors</b>. Modeling Errors cannot be avoided.
If one is able to write down an exact formula without errors, one can anyway
in general only be able to do calculations with round-off errors.

Subsection 6.2. Goals

The discription of Goals is analogous to that
 for Properties. 
As for Properties, a basic remark is that in order to discuss Hierarchies
efficiently, it is useful if the Goals are described
numerically. So we will assume that each GL(U) consists
of a finite list of numbers.

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Another basic remark about Goals, similar as for Properties, is that they are
not precisely <b> knowable</b>. There are always errors which
are in the nature of things. Hence, the list of numerical
values for Goals should also include an estimate of the
precision of these values, such as the number of accurate
digits.

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If the unit V has Superior, Sup(V)=U,
 then the Goals GL(V) come
from the Goals GL(U).
Since the Goals GL(U) have errors, these errors
transmit to errors in GL(V). To have an efficient description
of the Hierarchy, it is useful also to describe the
dependence of GL(V) on the Goals of its superior
in terms of functions,<br>
GL(V)= T(GL(U))<br>



In writing down the function T, one usually will make simplifying
assumptions so as to make calculations more feasible. Hence one
introduces again <b> Modeling Errors</b>. Modeling errors cannot be avoided.
If one is able to write down an exact formula without errors, one can in
general anyway only be able to do calculations with large round-off errors.

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Subsection 6.3. States

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In order to discuss Hierarchies
efficiently, it is also useful if the States are described
numerically. So we will assume that each St(U) consists
of a finite list of numbers.

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As for Properties and Goals, States are
not precisely <b> knowable</b>. There are always errors which
are in the nature of things. Hence, the list of numerical
values for States should include an estimate of the
precision of these values, such as the number of accurate
digits.

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Let a Group consist of Units V1,...,Vn.
Instead of considering the list of States
St(V1),...,St(Vn) (which might be a rather
large list of numbers), it is often
more manageable to consider a shorter list
which captures the main features of the States
St(V1,...,St(Vn. We will call
this the State of V1,...,Vn and denote
it by St(V1,...,Vn). This introduces Modeling
errors, but as before, if we instead use the
whole list of States, the Calculations are anyhow
usually only doable with round-off errors.
The State St(V1,...,Vn will be a point
in a finite-dimensional space  F, usually
Real, Complex or Projective m-dimensional space, for
a small dimension m.

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Section 7. Time

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Hierarchies are highly unstable. In this section we
discuss how they vary with time.

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Subsection 7.1. Structure of Hierarchies

In this subsection we discuss Hierarchies  H and
their composition of Units and some ways how this changes over time.

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Hierarchies can be <b> created</b>, i.e. a collection of Units
can be assembled and organized in a Hierarchical fashion
as Superiors and Subordinates. Opposite, a Hierarchy can also
<b>cease to exist</b>.

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Let U be a Unit (but not the top Unit) in a Hierarchy. Then
 U is the top Unit of a
<b> Subhierarchy </b>  H|U, consisting of U,
 Sub(U), Sub(Sub(U)), ... .
We say that the Subhierarchy is <b> closed </b>  if H|U
ceases to exist. Then the rest of the Hierarchy, consisting of all
the remaining Units,  H\ H|U
has the obvious hierarchical structure.
Opposite, given two Hierarchies,  H1, H2
we say that H1 is greffed on H2 if 
the top Unit of H1 is added as a Subordinate Unit
of some Unit in  H2. In that case  H= H1
and H2 has a natural structure of a Hierarchy.
Also, Hierarchies  H1,..., Hn can be
 joined if a new Hierarchy  H is created by
 choosing a new top Unit U0
whose Subordinates are the top Units of H1,...,  Hn.
Then H= U0 and H1 and ... Hn.
The opposite of Hierarchies joining is that a Hierarchy  splits, i.e.
the top Unit is removed and each of the Subordinates V1,...,Vn becomes
 the top Unit of its own Hierarchy,  H|Vi.

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A Hierarchy can be refined, i.e. a subset W of the Subordinates
 of some Unit U might be organized in Groups G1,..., Gm and
each Group Gi gets a new Superior Vi. Then U becomes the Superior
of the Vi as well as the remaining Subordinates not in W. The opposite
of refinement is  simplification. In that case 
 some of the Subordinates of a Unit U are removed and their Subordinates
become Subordinates of U.

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Subsection 7.2. Properties

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When a Hierarchy changes with time, the Properties of the Units
also change. So in general Properties of Units, P(U) vary
with time, P(U,t).

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Subsection 7.3. Goals

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When a Hierarchy changes with time, the Goals of the Units
also change. So in general Goals of Units, GL(U) vary
with time, GL(U,t).

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Subsection 7.4. States

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Let U be a Unit with Subordinates V1,...,Vn. We represent the
State of V1,...,Vn as a point in the State space F.
The State will depend on time. We get a map T: F to F<br>

St(V1,...,Vn,t+1)= F(St(V1,...,Vn,t),Q)<br>



expresses the State at time t+1 as a function of the State at time t.
Here Q refers to other relevant variables, such as the States of the
 Subordinates of the Vi and of their Superior U, as well as the
 Properties and Goals of the Vi.

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Section 8. Several Hierarchies

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Often several Hierarchies interact. For example, two
Hiearchies  H1, H2 might Cooperate
or Compete.
Also, there might be one main Hierarchy H
and the Units in H might be Units which
also belong to other Hierachies, for example from two
Competing Hierarchies.









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