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Formal Hierarchical System

Hierarchical System


  1. A hierarchy is one dimensionally oriented but may represent an inifite-dimensional structure or space of structures.

  2. Each hierarchy is coupled with one inverse (inverted) hierarchy having exactly the same structure with reversible orientation.

  3. A contextual difference is made between elements and node of a hierarchy. The nodes of a hierarchy according to this generilez conception may have both multiple outgoing and ingoing connections.

  4. A hierarchy need not be connected in all parts. The mutual belonging of several parts to one and the same structure is predetermined by context and poses individual coinciding characteristics, stating that a number of (dis)connected structures are (parts of) one and the same hierarchy; that is consequence of its contextual characteristics. The latter are called invariant characteristics.

  5. The contextual invariant characteristics are defined in natural language; their variety is unrestricted.

  6. Mutations” and “recombinations” are performed by:

    operations, which are similar to the arithmetic ones and actually build an algebra in the special matematical sense,

    altering the values of meaning or,

    some special operations typical only of this object of investigation and bound to the context of the hierarchy

  7. Deadlocks are possible, but are then acceptable only if they can represented a interuptible at a defined spot under defined conditions.

invariant characteristic

Processes defined as class of hierarchies

Description in natural language


time axis

usually top-bottom; by including the feed-back to every connection as an object within the subsystem the inverse direction and inverse hierarchy are predefined


sequencing events according some rule(s) or some natural law(s)

from the whole to its parts according to the specific function(s) actually entirety of connections


situations that possess some common features and properties or are summoned in one set by some law

no difference is made between elements and nodes


events, occuring after, following or evolving from one another

subsystems and elements; the latter are actually the end nodes


one step of evolution between one event and its following

subordination comprising the objects’ entry point, process exit feedback, and restrictions, as well as their properties in a conjoint systemic process

algorithm of decomposition

sequencing events, disclosing their new or changed properties, notifying the moment or time interval when they take place and how long they (might) last

sequencing the processes within the subsystems according to their subordination and support ensuring the results


possible: 2 events may be followed by one and the same

non per definition; counterexample can be demonstrated


possible: the sequencying may follow several rules at a time

none by definition; counterexamples are the multitude of possible functions to which the multitude of measures are applied, the goal hierarchy of restrincting criteria


1) time and space units

2) units for measuring properties of a situation: light, color or warmth intensity, spectral values, velocity, weight, height, temperature, etc.

a great multitude of physical, chemical, etc. measures, as well as specific control criteria; fuzzy methods of measurement play an increasing role.

Source: nfm

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out. invariants

von Neumannutility*