Universal hypergraphic automata representation by autonomous input symbols

Hypergraphic automata are automata with state sets and input symbol sets being hypergraphs which are invariant under actions of transition and output functions. Universally attracting objects of a category of hypergraphic automata are automata $\mathrm{Atm}(H_1 ,H_2)$. Here, $H_1$ is a state hypergraph, $H_2$ is classified as an output symbol hypergraph, and $S=\mathrm{End} H_1\times \mathrm{Hom}(H_1,H_2)$ is an input symbol semigroup. Such automata are called universal hypergraphic automata. The input symbol semigroup $S$ of such an automaton $\mathrm{Atm}(H_1 ,H_2)$ is an algebra of mappings for such an automaton. Semigroup properties are interconnected with properties of the algebraic structure of the automaton. Thus, we can study universal hypergraphic automata with the help of their input symbol semigroups. In this paper, we investigated a representation problem of universal hypergraphic automata in their input symbol semigroup. The main result of the current study describes a universal hypergraphic automaton as a multiple-set algebraic structure canonically constructed from autonomous input automaton symbols. Such a structure is one of the major tools for proving relatively elementary definability of considered universal hypergraphic automata in a class of semigroups in order to analyze interrelation of elementary characteristics of universal hypergraphic automata and their input symbol semigroups. The main result of the paper is the solution of this problem for universal hypergraphic automata for effective hypergraphs with $p$-definable edges. It is an important class of automata because such an algebraic structure variety includes automata with state sets and output symbol sets represented by projective or affine planes, along with automata with state sets and output symbol sets divided into equivalence classes. The article is published in the authors' wording.