Aleksander K. Cherkashin

V.B. Sochava institute of geography SB RAS

This study aims are to create meta-models of the system organization with different complexity and a mathematical description of representative capacities that ensure paired subsystems interactions. To solve the problem, the basic ideas of A.A.Bogdanov’ general organizational science are involved, where an organization is an ordered set of internal relationships and system properties with a certain functioning mode. The mathematical foundations of systems theory are also used, namely, set theory, category theory, and differential geometry of the fiber bundle of spaces on a manifold of connections of variables. At various abstract levels, normative meta-models as polysystem without local feedback of coordination management are created in the form of commutative diagrams of the interaction of monosystems, taking into account external or internal factors of influence and possible changes in development goals. For metamodeling of various applications, existing software is used that supports any modeling notation, or traditional methods of scientific research are used, when many models of different objects are made based on a single meta-model. A meta–model is formed as a composition of classes, attributes and class relationships, and is formalized as a generalized organizational function – a manifold as bundle base for distinguishing a polysystem of classes, for example, in the form of separation of powers, work, and the powers of managers and executors. The communication interface between two classes expresses a relationship where functional changes in one class directly or through representatives affect the other class. Metamodeling notation in terms of category theory and mathematical analysis make it possible to reduce the abstraction of expressions in the general theory of systems and structural diagrams, and to derive calculation formulas. Through the fiber bundle procedure, the decomposition of the organizational function into three acts of entry, exit and monosystem state transformation in the form of a universal bilinear function of relative variables is justified, which describes the change of monosystems and the performance of representative actions. According to the limitations on this function, the concentric structure of each layer (fiber) "center-core-periphery" is axiomatically defined and the mechanism of polysystem connectivity of the fibers is modeled. The presented formal patterns and their graphical schemes mathematically display the structure and function of metamodels of organizational systems in conceptual and analytical terms and allow them to be applied in the analysis of data and knowledge, symbolically describe previously empirically established rules of self-organization in nature and society. The development of a bilinear description of an organizational function in parts through a tangent bundle is its quadratic representation in the form of a field of pairwise competitive interaction of objects.

organizational activity, fiber bundle procedure, normative polysystem, representative functions, categorical notation of metamodeling, commutative diagrams, organizational fields, universal function, mathematical meta-model