"The dynamic of product potential social systems and representation theory"
The product potential system is a locally interacting system with a potential (defined by requiring that a product-integral along any closed path on a graph equals to an identical transformation) system of psychological reactions (consisting of marks or fields on the edges of the graph of relations). A locally interacting process for the product potential system of relations can be given by an algebraic representation of an process of multiplication on the randomly chosen so call “control matrix”. We found one to one maps between thermodynamic states of system (the thermodynamic state for the system is measure) and so call “left ideals” on the semi-group of control matrices. The ideal matrices have a very important property: when an arbitrary stochastic/control matrix is multiplied from the left by an ideal matrix one obtains a left ideal matrix. So the set of left ideal matrices is the termination set for our stochastic product process (spatial Markov’s chain). It means that once the system reaches the termination set the process can never leave the termination set. Thus the left ideal matrices play a crucial role in the description of our process.