On problem of abstract definability of universal hypergraphic automata by input symbol semigroup

Hypergraphic automata are automata, state sets and output symbol sets of which are hypergraphs, being invariant under actions of transition and output functions. Universally attracting objects in the category of such automata are called universal hypergraphic automata. The semigroups of input symbols of such automata are derivative algebras of mappings for such automata. Semigroup properties are interconnected with properties of the automaton. Therefore, we can study universal hypergraphic automata by investigation of their input symbol semigroups. In this paper, we solve a problem of abstract definability of such automata by their input symbol semigroups. This problem is to find the conditions of isomorphism of semigroups of input symbols of universal hypergraphic automata. The main result of the paper is the solving of this problem for universal hypergraphic automata over effective hypergraphs with $p$-definable edges. It is a wide and a very important class of automata because such algebraic systems contain automata whose state hypergraphs and output symbol hypergraphs are projective or affine planes. Also they include automata whose state hypergraphs and output symbol hypergraphs are divided into equivalence classes without singleton classes. In the current study, we proved that such automata were determined up to isomorphism by their input symbol semigroups and we described the structure of isomorphisms of such automata.