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All models have in common that they encode experience and always involve signs, signals, syntax, semantics and an ability to decode and derive meaning from what is encoded. Therefore, an ecological model can be considered a text T, written in a formal (mathematical) language defined by the authors as L(M_T).

The language chosen is a regular language generated by regular grammars which in turn are generated by finite automaton.

A text results from a union of some finite automata defining a textual grammar. Recognoscitive and generative grammars of words, defining the basis of the L(M_T) language are developed.

A mathematical modelling of ecological systems is developed in parts I and II of this research project.

A language for ecological systems modelling has been defined and its implementation discussed.

The purpose of this paper is to present a further extension of the regular grammars of the language L(M_T).

This language may represent an ecological model, written in a formal mathematical structure.

The paper presents and develops recognoscitive and generative grammars of sentences (flow equations) together with a synonymy relationship, which characterize the L(M_T) language.

Provides a discussion of modelling of language.

Any mathematical model underlying a system of the complex reality (for example, in ecology, socioeconomy, etc.) defines a text written in a given language which belongs to a literature. A text‐model based language, L(M_T) is developed, which focuses on its syntax. Three generative grammars are built for its alphabet (lexic unities). Further, the linguistic entropy of this language is analyzed from the Vakar’s point of view. Finally, this theory is applied to a particular ecological model, as an example of a text written in L(M_T).

It is known that a mathematical ecological model and, in general, a particular methodology of modelling, can be considered a literary text written in a formal mathematical language. In this context, stylometric mathematical laws such as Zipf’s (range‐frequency and number‐frequency) can be applied to obtain information parameters in different semantic levels within the same model. Adapts several of these laws and introduces new elements, lexic units, operating and separating units, to carry out several statistical analyses upon two models or texts. The estimated slopes in the regression equations obtained in the present work are compared with the results of previous papers where Mandelbrot’s law was applied and comparisons between them are shown.

The purpose of this paper is to demonstrate mathematically the impossibility of achieving a utopian society. Demonstrate that any attempt to correct deviations from a hypothetical trajectory whose ultimate goal is the utopia, increasingly demands more work, including measures that lead to terror, which may even be absolute, leading to the horrible paradox that in seeking paradise hell is constructed.

Scientific tools that the authors have used are: the theory of the system linkage, alysidal algebra, kinematic theory and vector analysis.

Myths are the substrate of some complex systems of beliefs and utopia is its ultimate goal. The use of the combination of the theory of trajectories, belonging to the alysidal algebra, the theorem of unintended effects and kinematics theory provides an approximation to deviations suffering utopian ideological currents and their corrections.

This paper is a continuation of other previous papers developing the theory of complex societies.

The purpose of this paper is to provide an introduction to alysidal algebra. Deontical impure systems are systems whose object set is formed by an s‐impure set, whose elements are perceptuales significances (relative beings) of material and/or energetic objects (absolute beings) and whose relational set is freeways of relations, formed by sheaves of relations going in two‐way directions. Objects and freeways form chains.

The paper looks at the mathematical and logical development of human society structure.

Existence of relations with positive imperative modality (obligation) would constitute the skeleton of the system. Negative imperative modality (prohibition) would be the immunological system of protection of the system. Modality permission the muscular system, that gives the necessary flexibility to the system, in as much to the modality faculty its neurocerebral system, because it allows him to make decisions. Transactions of energy, money, merchandise, population, etc. would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, on the existence of a legislative body with their obligations, prohibitions and permissions that regulate them.

The concepts of alysidal set are introduced, whose elements are chains and coupling function between alysidal sets. Environment is formed as well by different DIS. That is to say, by an alysidal set whose elements are simultaneously systems. Specific interchanges (stimulus‐response) leave certain nodes and act on certain nodes of the alysidal sets (stimulus environment‐DIS‐response environment). The paper defines a special coupling function, denominated gnorpsic function, that can be used for algebraic operations between alysidal sets.

Deontical impure systems are systems whose object set is formed by an s‐impure set, whose elements are perceptuales significances (relative beings) of material and/or energetic objects (absolute beings) and whose relational set is freeways of relations, formed by sheaves of relations going in two‐way directions and at least one of its relations has deontical properties such as permission, prohibition, obligation and faculty.

The paper looks at the mathematical and logical development of human society structure.

Existence of relations with positive imperative modality (obligation) would constitute the skeleton of the system. Negative imperative modality (prohibition) would be the immunological system of protection of the system. Modality permission the muscular system, that gives the necessary flexibility to the system, in as much to the modality faculty its neurocerebral system, because it allows him to make decisions. Transactions of energy, money, merchandise, population, etc. would be the equivalent one to the sanguineous system. These economic transactions and inferential relations, depend, as well, of the existence of a legislative body with their obligations, prohibitions and permissions that regulate them. A Social System Σ may be considered like an alysidal set with an only alysidal element. The authors consider two theories: Enlarged theory and Reduced theory.

This paper is a continuation of two previous papers – Part I published in Kybernetes, Vol. 41 No. 1/2 and Part II published in Vol. 41 No. 5/6 – and develops the theory of deontical impure systems.

This paper aims to refer to a subjective approach to a type of complex system: human ecosystems, referred to as deontical impure systems (DIS) to capture a set of properties fundamental to the distinction between human and natural ecosystems. There are four main phenomenological components: directionality, intensity, connection energy and volume. The paper establishes thermodynamics of deontical systems based on the Law of Zipf and the temperature of information.

Mathematical and logical development of human society structure.

A fundamental question in this approach to DIS is the intensity or forces of a relation. Concepts are introduced as the system volume and propose a system thermodynamic theory. It hints at the possibility of adapting the fractal theory by introducing the fractal dimension of the system.

This paper is a continuation of other previous papers and developing the theory of DIS.

The authors continue developing a Linguistic Theory of the Ecological Systems, but in terms of the Semantics. Each symbol (transformed function) is syntactically a lexeme, carrying an associate sememe or atomic semantic unit. Each sememe can be decomposed in semes or quantic semantic unities. They are studied and semantic systems, associated with the syntactic, that serve them as suprastructure, in terms of two levels: the quantic and the atomic. Also, it is demonstrated that for all models of the complex reality, there will always exist a most complex model from the syntactic and semantic point of view.

Mathematical models are constructed at the interface between practice, experience and theories. The function of models puts us on guard against the privilege granted to what is accepted as abstract and formal, and at the same time puts us on guard against a static and phenomenological conception of knowledge. The epistemology of models does not suppress in any way the objectives of science: only, a dogmatic conception concerning truth is removed, and dynamic and dialectical aspects of monitoring are stressed to establish the most viable model. The purpose of this paper is to examine hybrid methodologies (inductive-deductive) that may either propose hypothetical causal relations and seek support for them in field data or detect causal relations in field data and propose hypotheses for the relations detected.

The authors follow a dialectical analysis for a type of inductive-deductive model.

In this work, the authors present an inductive-deductive methodology whose practical result satisfies the Hegelian dialectic. The consequent implication of their mutual reciprocal integration produces abstractions from the concrete that enable thought. The real problem in this case is a given ontological system or reality.

The essential elements of the models – variables, equations, simulation and feedback – are studied using a dialectic Hegelian theory.