Relative elementary definability of the class of universal graphic semiautomata in the class of semigroups

The theory of automata is one of the branches of mathematical cybernetics, which studies information transformation devices that arise in many applied problems. In this paper, we study automata without output signals and call them semiautomata. Depending on specific problem, semiautomata are considered, in which the set of states is equipped with an additional mathematical structure consistent with the transition function of a semiautomaton. In this paper, we investigate semiautomata over graphs (what is known as graphic semiautomata), the set of states of which is equipped with the mathematical structure of a graph.
Universal graphic semiautomaton $\text{Atm}(G)$ is the universally attracting object in the category of semiautomata, for which the set of states is equipped with the structure of a graph $G$, preserved by the transition function of the semiautomaton. The input signal semigroup of such semiautomaton is $S(G) = \text{End}\ G$. In this paper, we consider the problem of the relatively elementary definability of the class of universal graphic semiautomata over reflexive quasi-acyclic graphs in the class of semigroups, and applications of the obtained relatively elementary definability.