Discrete automata on semilattices

The theory of discrete automata on semilattices is one of the important achivements of the Tomsk State University scientific School on applied discrete mathematics. It is a comparatively young branch of the science connecting mathematical cybernetics and abstract algeabras. Inside this branch, we have firstly succeeded in formal defining such notions related to the discrete control systems as the dynamic behaviour, the hardware realizability, the adequate model and its accuracy, and also in solving the problems of logical synthesizing such systems with a given dynamic behaviour and with the possibility of hardware realization at the transistor level and of modelling their dynamic behaviour adequately with any given accuracy. The paper is presented on behalf of 50 years jubilee of the School. It is an extended abstract of the same name monograph by the author issued at TSU in 1993 and now being out of access. All the main results obtained in the theory of discrete automata up to that time are presented in the paper.