International Encyclopedia of Systems and Cybernetics

Prof. M. KALAIDJIEVA (Institute of Control and Systems Research of Sofia, Bulgaria) proposed the following postulates for a general concept of hierarchy. She writes:

POSTULATES OF GENERAL HIERARCHIE

1. A general hierarchie GH is an one-dimensionally orientated multi-dimensional structure of constructs (structures), up to infinite dimensional.

2. Each general hierarchie is represented by two coupled GHs – one GH with an inverse general hierarchie having exactly the same structure, but defined with the opposite orientation of its direction.

3. Deadlocks are tolerable within a GH, only if they can be represented as cycles interruptible at a defined place under defined conditions.

4. A theoretical contextual difference is made between elements and nodes of a hierarchie. Many methods of modeling GHs rely on this seperation.

5. The nodes of a GH may have both multiple outgoing and in-going connections. This particular feature is polyhierarchie. A general hierarchie having the feature of polyhierarchie is called polyhierarchie GH or shortly – polyhierarchie.

6. A general hierarchie need not be connected in all its parts. The mutual belonging of several parts possess individual identic or similar contextual characteristics, stating that number of (dis)connected structures are (parts of) one and the same general hierarchie.

7. The contextual characteristic is called invariant characteristic (IC). The contextual ICs are described (defined) in natural language; their variety is unrestricted. There are nine primary ICs: direction, aspect of decomposition, element, node, connection, (algorithm of, rules of decomposition, polihierarchie, polythematics, measure. They might be independent of one another or not depending on the context of GH design and analysis. Four secondary ICs are derived from the system of the primary nine composition, inverse hierarchie, level and step.

8. General hierarchy (ies) may be modified (rearranged) by

a) Operations wich build by construction set up an algebra in the special mathematical sense of this term. They are addition, subtraction multiplication and division. Basic elements of the algebra are of two kinds of bases:

- 1 order base of algebra containing constructs of 2 nodes bound with 1 connection (or 2 connections bound with 1 node);

- 2 order base of the algebra containing constructs of 1 nodes, all its subnodes and bound with as many connections.

b) Altering the values or meanings of invariant characteristic(s). Such changes may also change the degree of abstractness of modeling complex objects in GH(s) or change the context of modelled constructs, e.g. for comaparative study.

c) Special operations typical only for the set of general hierarchies and bound to the context of the GH(s): normalize GH(s) on one and the same level, haul up and haul down across level (steps).