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TitleTopics in the General Theory of Structures
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General Editors: W. Leinfellner and G. Eberlein

Series A: Philosophy and Methodology of the Social Sciences

Editors: W. Leinfellner (Technical University of Vienna)
G. Eberlein (Technical University of Munich)

Series B: Mathematical and Statistical Methods

Editor: H. Skala (University of Paderborn)

Series C: Game Theory, Mathematical Programming and Mathematical

Editor: S. Tijs (University of Nijmegen)

Series D: System Theory, Knowledge Engineering and Problem Solving

Editor: W. Janko (University of Vienna)


Editor: W. Janko (Vienna)

Editorial Board

G. Feichtinger (Vienna), H. T. Nguyen (Las Cruces), N. B. Nicolau (Palma de Mallorca),
o. Opitz (Augsburg), H. J. Skala (Paderborn), M. Sugeno (Yokohama).


This series focuses on the design and description of organisations and systems with application
to the social sciences. Formal treatment of the subjects is encouraged. Systems theory,
information systems, system analysis, interrelated structures, program systems and expert
systems are considered to be a theme within the series. The fundamental basics of such concepts
including computational and algorithmic aspects and the investigation of the empirical
behaviour of systems and organisations will be an essential part of this library. The study of
problems related to the interface of systems andorga'nisations to their environment is
supported. Interdisciplinary considerations are welcome. The publication of recent and
original results will be favoured.

Page 104


When the set X is denumerable and when the subset X'~ C X of elements
x c X, for which Cf (x) '" x is finite. we denote by Xf set ~U<f'(X''f)' where.
as usual. ce (X'le) indicates set [ 'f (x) I x f. XI(}. Evidently. set X ~
is finite and functionf is a correspondence of set X~ to itself. There-
fore. a finite graph with the above-mentioned connected components corresponds
to function ~ (x).

The function f (x) is called the subordination function or function S.
Oriented graphs with numbered vertices corresponding to functions S

will be called CH-graphs.
Thus. function S defines a finite CH-graph using the set of vertices

X'f . To all the other elements x E. X'\ X l(l (whose number may be infinite)
correspond isolated vertices. which will bE! considered "external to the
structure" and will not be represented in a figure.

Suppose, now. that set X could be represented as a partition into k mi-
nimal subsets X~ ,which makes

Xlf = U X~

ce (X~ ) ~
correspond to itself.

(for i '" j f/J). i 1. 2 ••.•• , k

Then. each of these subsets supplies vertices with the i-th connected
component of· the CH-graph.

Due to the fact that functions S have a single value. it follows that
only a cycle in the i-th connected component corresponds, on graphrf • to
subset X~ f X1e, such that ~. (~;i ) = X~i • If there. is an element xn £. X~i
such that 'f (~) = xn ' then XI/ = £ xnl and subset X~ should not include any
other element satisfying this equality. On graphr~ a tree. whose root is an
encircled vertex, corresponds to such a subset.

For X~i we will use the term complex cycle of the i-th subset ~ , and
for elements x t X~ belonging to complex cycles. the term complex elements
of graph T'f' •

The structure of a CH-graph is univocally determined if. for a given
function S, its subset X~ and the partition of this subset into cycles x~i
is found and, further. it is established which of these are isolated.

To determine subset X~ it is sufficient to consider a sequence of sets
XO' Xl' X2'·······. Xk .·····, where Xo = X'f'. Xi +1 = If (Xi)' i = 1, 2 ••..•

A subset X~ is a set Xk. therefore ~(Xk) = Xk • In reality. graph
r ';+1 with vertices taken from set Xk+1 coincides with graph r ~ with

vertices taken from set Xk • without the hanging vertices. If a CH-graph
contains a chain with maximal length k. having no edges common with any of
its cycles. then X~ = x'ie and X* ~ X'f • To distinguish all cycles X*,) of
subset X~ , which is the partition of X~ into minimal subsets. for which

. *. *.
~(X~1 ) = X~ , requires. in general, a complete analysis of all the

subsets in X~ • To facilitate this search. a procedure can be proposed.
Let be x.t X~and form ~ chain X<fi = [%(X1)' 'f !~X1)' ..•• ' cpk(X1)]

for which Cf 1(x1) = 0/ ( <f 1-1(x1» and f s+1(x1) E: xc( .
This chain always exists and belongs to the j-th connected component.
Now, let X 2 E X'f \ (X~ u X~) and form, for x2' the same chain ~2
This chain will not have intersections with the first one and will re-

present the second cycle of the CH-graph. By continuing this procedure, the
partition of subsets into cycles will be obtained.

Page 105


Consider, now, the procedure for partitioning the whole set Xo/ of the

CH-graph into subsets xf· 1
For. xi f XL( \ X*'( w~ can build a chain X~ = fXl' Cf l(xl)' •.. ' f'S(xl)j

with f 1(xl) ( f ( ~ 1-l(xl» and ~ s+l(xl) E X~J Such a chain
always exists and belongs to the j-th connected component. Now, for

x2 E. X~ \ x;f U X~ let us construct a chain xq = {x!, Cf.l(x~~, •.. ~t(X2)S
with 'f. t+l(x2) E X~ u X~ . If, at the same time, f t+ (x2) E. X~ U Xtf
all the elements of this chain also belong tn the i-th connected component.
Otherwise, Cf t+l(x2) £; ~i and therefore, all these elements belong to the
i-th connected component.

The continuation of such a procedure leads to the partition of set X
into its component subsets.

Example 2.1. Let x be a set formed by the first 20 letters of the latin
alphabet: X = a, b, c, ••...•. ,s, t. Using X, two functions Cf 1 and tf' 2 are
defined by Table 2.1

x a b c d e f g h i j k 1 m n 0 p q r s

If I (x) b a c d p f a a g j m 1 b n m a q r m

Cf 2 (x) h s c 1 1 P b f 1 0 b 0 c s c a 0 n s


Subsets xt for these functions are

xllf I [ a, b, c, g, h, i, k, m, 0, p, s}

xlCf 2 { a, b, d, c, f, g, h, i, j, k, 1, m, n, 0, p, q, r, t J
Functions !f I' <f2 make these subsets correspond to subsets

'f l(XI Cf 1) = ta , b, g, m, p}
Cf 2tXl If 2) = {a, b, c, f, h, n, 1, 0 p, s J

Subsets X'f for both correspondences are

x 'f 1 = Xl ~ I U 'f 1 (Xl ce I) = Xl Cf I

X if 2 = Xl ce 2 Uf(! 2(X1 <t' 2) = X





* ,~ In order to distinguish subsets X ~ 1 and X If 2' construct the sequence
of subsets xO r l' Xl Cf I'···· and X°'f 2' X1'f 2'.... as follows:
X0'fl =X~l; Xl'fl = la, b, g, m, pl; x2rl = [a, bJ; x3<f'1 =x2'f\
xOIf 2 = X?'2; Xlr 2 = fa, b, c, f, h, 1, n, 0, p, s}

X21f1 =£a, c, f, h, 0, p, s}; X1'2 = [a, c, f, h, p, s1; X4r2 =X3f2;

Page 208


C-Calculus 163, 174, 184-185
C-Calculus, a measure theory model 177-180
C-Calculus mathematical framework 175-176
C-Contour extraction 189
C-Contraction 191
C-Dilation 192
C-Filter 185-186
Classes of Function S 76-85
Coins Distribution Function 24-25
Coins Invariance principle 24-25
Collective Choice 140-142
Collective Opinions 140
Complex Hierarchical Graphs (CH-Graphs) 97-99
CH-Graphs Hultiplication 101-107
CH-Graphs Structure 98-101
Convergence and filtering in C-calculus 166-170
C-Product 164, 184
Cramer-Rao inequality 204
Cross-entropy 201
C-Separation 190
C-Set 164, 184
C-Sum 164, 184

Distance 72
Dynamics of trees 69-72

Function S, First order equations 86-93
Function S Graph 74-75
Function S Higher order equations 94-96
Function S Subordination 72-73

Graph Dynamics Problems 129-131
Graph Trajectory 71, 10t~106

Heisenberg inequality 204
Hierarchical Modular Systems (HHS) 11, 13-14,61
HMS-Distribution Law 15-17
Hierarchical Systems 9-10, 61
Hypergraph in Structural Properties 152-155

Individual Choice 138-139

J-Divergence 202

Page 209


Levels, Formation of 12-13
Levels, Information of 12

Majority 137
Monetary Systems 18-23
Monetary System reorganization 28
Movement Analysis 193

Natural Language, Self-organizing system 60-64

Partition 9
Pattern 5-6
Population Distribution on a Territory 29-35
Predicate Calculus 149-152

.Quantification 6-8

Refinement 11, 13-14

Set 5
Shannon Entropy 200
Stability 8
Structure 5
Structure and Stability of a complex envelope 113-117
Subsets of periodic solution 107-112
System 5, 149

Tournament Functions 157
Tournament Matrix 158

WEB Grammars 122-128


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