On structure of isomorphisms of universal graphic automata

Automata theory is one of the branches of mathematical cybernetics, that studies information transducers that arise in many applied problems. The major objective of automata theory is to develop methods by which one can describe and analyze the dynamic behavior of discrete systems. Depending on study tasks, automata are considered, for which the set of states and the set of output signals are equipped with additional mathematical structure preserved by transition and output functions of automata. We investigate automata over graphs and call them graphic automata. Universal graphic automaton $\mathrm{Atm}(G,H)$ is a universally attractive  object in the category of such automata. The semigroup of input signals of the automaton is $S(G,H) = \mathrm{End}\ G \times \mathrm{Hom}(G,H)$. It can be considered as a derivative algebraic system of the mathematical object $\mathrm{Atm}(G,H)$, which contains useful information about the initial automaton. It is common knowledge that properties of the semigroup are interconnected with properties of the algebraic structure of the automaton. Hence, it is possible to study universal graphic automata by researching their input signal semigroups. Earlier the authors proved that a wide class of such kind of automata are determined up to isomorphism by their input signal semigroups. In this paper, we investigate a connection between isomorphisms of universal graphic automata and isomorphisms of their components — semigroups of input signals and graphs of states and output signals.

automata, graph, semigroup, isomorphism, automorphism