Abstract Systems Theory

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2020 Mathematics Subject Classification: Primary: 93A10 [MSN][ZBL]

This article discusses systems in the context of general systems theory. For systems in the sense of logics, see formal systems.

History and Motivation

In many sciences, e.g. sociology, biology, cybernetics, chemistry, politics, economics (see the Wikipedia article Systems theory), the notion of a system was defined independently from each other. Thus, problems were inavoidable as soon as topics were discussed at the interplay between them (e.g. cybernetic models of biological systems). A more fundamental definition of a system was required encompassing the different concepts.

These definitions have in common that a system consists of elements related to each other [S]. Accordingly, Hall and Fagen proposed a definition of a general system as 'a set of objects together with relationships between the objects and between their attributes' [HF],[MP] in 1956. Their paper was published in the first volume of the Journal General Systems. Ludwig von Bertalanffy, one of the editors of the Journal, uses a similar definition in his famous book General Sytem Theory [vB] in 1968. The formalization of this view of a system as a relation between two or more sets was subject to others, however, as for example Mesarovic and Takahara [MT1],[MT2]. We will follow their approach here, but not without mentioning that a number of other definitions of an abstract system have been proposed (Klir [K1],[K2],[K3],[K4], Lin [L], Polderman and Willems [PW],[Wi], Rosen [R1],[R2], Wymore [W1],[W2], Wang [Wa]). Their basic ideas may correspond to the definition of Mesarovic and Takahara in essence, but their theories on the whole usually differ considerably. A standard definition accepted by the whole systems community seems to be still missing.


A system is defined as a relation $S\subseteq I \times O$, whereby $I$ and $O$ are sets representing inputs and outputs [MT2]. Consequently, $S$ will be called an input-output (or elementary) system more specifically. This definition reflects some kind of black-box view on the system, since the internal structure or function is not represented. It deals only with the correlations between inputs and outputs.

Sometimes, more than two factors are considered and a system is defined as $S\subseteq \prod_{i\in J} F_i$. Therein, the sets $F_i$ are the objects belonging to the system. A set $F_i$ gives the totality of different properties, which this object may potentially have [MP]. This version of a systems definition reveals some insight into the internal structure and function of the system. It is used for describing composed (or nonelementary) systems.

Elementary and Nonelementary Systems

Both versions of a systems definition given above are interrelated with each other. Two or more elementary systems can be combined to a nonelementary system. In this way a hierarchy of systems can be built. A goal-seeking system $S \subseteq I \times O$ is a simple example for a nonelementary system. It is composed of two elementary systems $S_1\subseteq (I\times O) \times M$ and $S_2\subseteq (M\times I) \times O$. The subsystem $S_1$ gives admissible goals $m\in M$ depending on the inputs $i\in I$ and outputs $o\in O$ of $S$; the subsystem $S_2$ on the other hand defines a relationship between input $i\in I$ and goals $m\in M$ with appropriate outputs $o\in O$.

Constructing Systems

The above definition of an abstract system is general enough to be used in most applications. On the contrary, it is too general for the development of a rich theory with many nontrivial properties. Thus, abstract systems may be extended by additional structures. Typical structures are algebras (e.g. linear systems), function spaces (e.g. time systems, dynamical systems), probability spaces (stochastic systems), ordering relations and so on. In some cases, such structures are used for describing the system class under consideration constructively. Žampa et. al. [ZSV] is following this approach for example. Their starting point are time-discrete systems. Systems with a continuous time space are introduced as limits of sequences of time-discrete systems with increasingly higher time resolution.

Special classes of Systems

Functional System
A functional System $S\subseteq I \times O$ is a system with $S$ being a function $S\colon I\rightarrow O$.
A Time System $S$ is a system, in which inputs and outputs are functions defined on a set $T$, i.e. $S\subseteq A^T \times B^T$. The set $T$ has to be equipped with a total ordering relation and represents the time. Usually, time is formalized using stronger assumptions, e.g. by demanding the structure of a linear space as well. The weaker assumption used here allows to include e.g. discrete event systems, however.
Linear System
A system $S\subseteq I \times O$ is called linear, if both $I$ and $O$ are $K$-vector spaces and if $S$ is closed under linear operations: $$\begin{array}{rcl} s,s'\in S &\Longrightarrow & s+s'\in S\\ s\in S, \alpha\in K&\Longrightarrow & \alpha s\in S \end{array}$$

Morphisms between systems

Let $S\subseteq I \times O$ and $S'\subseteq I' \times O'$ be two systems. A system morphism $h\colon S\rightarrow S'$ (in the relational sense) is a pair $h=(h_I, h_O)$ of mappings $h_I\colon I\rightarrow I'$, $h_O\colon O\rightarrow O'$ fulfilling $(i,o)\in S \Longrightarrow (h_I(i),h_O(o))\in S'$. For functional systems, this definition of a morphism turns out to be inappropriate because the function property of $S$ is not preserved under the morphism. This has led to the notion of a system morphism $h\colon S\rightarrow S'$ in the functional sense; here, $h$ is a pair $h=(h_I, h_O)$ of mappings $h_I\colon I\rightarrow I'$, $h_O\colon O'\rightarrow O$ fulfilling $S'(h_I(i))= h_O(S(i))$ for $i\in I$.


[HF] A.D. Hall, R.E. Fagen, "Definition of a system", General Systems 1(1956)18-28
[K1] G. Klir, "An approach to General systems theory", Van Nostrand 1969
[K2] G. Klir, "Trends in General Systems Theory", Wiley, 1972
[K3] G. Klir, "Architecture of Systems Problem-solving", Plenum Press 1985
[K4] G. Klir, "Facets of Systems Science", Springer 1991
[L] Yi Lin, "General Systems Theory: A Mathematical Approach", Springer 1999
[M] M.D. Mesarovic, "On some mathematical results as properties of general systems", Mathematical systems Theory 2(1968)357-361
[MT1] M.D. Mesarovic, Y. Takahara, "Abstract Systems Theory", Springer 1989, LNCIS 116
[MT2] M.D. Mesarovic, Y. Takahara, "General Systems Theory: Mathematical Foundations", Academic Press 1975
[MP] G. Minati, E. Pessa, "Collective Beings", Springer 2006
[PW] J.W. Polderman and J.C. Willems, "Introduction to Mathematical Systems Theory: A Behavioral Approach", Springer 1998
[R1] R. Rosen, "Fundamentals of Measurement and representation of Natural Systems", North-Holland 1978
[R2] R. Rosen, "Anticipatory systems", Pergamon Press 1985
[S] J. Sanders, "Theoretical Approaches to Systems", Diploma Thesis, Bielefeld 2003
[vB] L. von Bertalanffy, "General Sytem Theory", George Braziller Inc. 1968
[Wa] Yingxu Wang, "Software Engineering Foundations: A Software Science Perspective", Auerbach Pubn 2007
[Wi] J. Willems, "Paradigms and puzzles in the theory of dynamical systems", IEEE Transaction on Automatic Control 36(1991)259-294
[W1] W. Wymore, "Model-Based Systems Engineering", CRC Press 1993
[W2] W. Wymore, "A Mathematical Theory of Systems Engineering, The elements", Wiley 1967
[ZSV] P. Žampa, P. Steska, K. Veselý;, "Multivariable linear discrete-time stochastic system continualization", WSEAS Trans. Syst. 3(2004)2898-2903
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