A. S. Fedulov

Now cognitive maps have found wide application for representation and the analysis of knowledge of difficultly formalizable application domain. Such tasks of the analysis of complex systems and problems as the analysis of dynamic behaviour, scenario analysis, generation of management decisions, can be solved by cognitive modeling. In this work fuzzy relational cognitive maps are considered. In this kind of cognitive maps the concepts take fuzzy value, and the influence between concepts is represented by fuzzy relations. Whole cognitive map can be represented on the basis of a matrix of fuzzy relations. Cognitive map is considered as the closed system and as system on which external influence can be sent. For this kind of cognitive maps the set of vector-matrix operations are offered. On the one hand, these operations allow calculating of transitive closure of a matrix of fuzzy relations. On the other hand, they allow simulation of fuzzy relational cognitive maps. Vector-matrix operations are formalized with the help of operations above fuzzy sets and relations. These operations are: multiplication of fuzzy sets and relations and aggregation of fuzzy sets and relations. For these operations minimally necessary algebraic requirements are received. These requirements consist in performance of some algebraic laws, such as associativity, commutativity, distributivity. For practical application it is necessary to express the operations above fuzzy sets and relations on the basis of operations above membership functions or α-level sets. One more problem which demands the decision is a problem of stability of transitive closure of the matrix of fuzzy relations. This problem is caused by expansion of uncertainty during fuzzy calculations. Еще одна проблема, решение которой рассмотрено в данной работе, это проблема устойчивости транзитивного замыкания матрицы нечетких отношений, вызванная увеличением степени неопределенности значений нечетких переменных в результате нечетких вычислений в процессе моделирования.