On definability of universal graphic automata by their input symbol semigroups

Universal graphic automaton $\mathrm{Atm}(G,G')$ is the universally attracting object in the category of automata, for which the set of states is equipped with the structure of a graph $G$ and the set of output symbols is equipped with the structure of a graph $G'$ preserved by transition and output functions of the automata. The input symbol semigroup of the automaton is $S(G,G')=\mathrm{End} G \times \mathrm{Hom}(G,G').$ It can be considered as a derivative algebraic system of the mathematical object $\mathrm{Atm}(G,G')$ 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, we can study universal graphic automata by researching their input symbol semigroups. For these semigroups it is interesting to study the problem of definability of universal graphic automata by their input symbol semigroups — under which conditions are the input symbol semigroups of universal graphic automata isomorphic. This is the subject we investigate in the present paper. The main result of our study states that the input symbol semigroups of universal graphic automata over reflexive graphs determine the initial automata up to isomorphism and duality of graphs if the state graphs of the automata contain an edge that does not belong to any cycle.