On definability of universal graphic automata by their input symbol semigroups

V. A. Molchanov ; R. A. Farakhutdinov

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 1, pp. 42-50.

Résumé

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.

Bibliographie

[1] Plotkin B. I., Groups of Automorphisms of Algebraic Systems, WoLters-Noordhoff Publ, Groningen, The Netherlands, 1972, 502 pp.

[2] Pinus A. G., “On the elementary equivalence of derived structures of free lattices”, Russian Math. (Iz. VUZ), 46:5 (2002), 42–45

[3] Pinus A. G., “Elementary Equivalence of Derived Structures of Free Semigroups”, Algebra and Logic, 43:6 (2004), 408–417 | DOI

[4] Gluskin L. M., “Semigroups and rings of endomorphisms of linear spaces”, Izv. Akad. Nauk SSSR. Ser. Mat., 23:6 (1959), 841–870 (in Russian)

[5] Gluskin L. M., “Semi-groups of isotone transformations”, Uspekhi Mat. Nauk, 16:5 (101) (1961), 157–162 (in Russian)

[6] Vazhenin Yu. M., “Ordered sets and their inf-endomorphisms”, Math. Notes, 7:3 (1970), 204–208 | DOI

[7] Vazhenin Yu. M., “The elementary definability and elementary characterizability of classes of reflexive graphs”, Izv. Vyssh. Uchebn. Zaved. Mat., 1972, no. 7, 3–11 (in Russian)

[8] Markov V. T., Mikhalev A. V., Skornyakov L. A., Tuganbaev A. A., “Rings of endomorphisms of modules and lattices of submodules”, J. Soviet Math., 31:3 (1985), 3005–3051 | DOI

[9] Ulam S. M., A Collection of Mathematical Problems, Interscience, New York, 1960, 150 pp.

[10] Plotkin B. I., Greenglaz L. Ja., Gvaramija A. A., Algebraic structures in automata and databases theory, World Scientific Publ. Co, River Edge, NJ, 1992, 296 pp.

[11] Molchanov V. A., “Semigroups of mappings on graphs”, Semigroup Forum, 27 (1983), 155–199 | DOI

[12] Molchanov V. A., Farakhutdinov R. A., “On universal graphic automata”, Computer Science and Information Technologies, Materials of the Int. Sci. Conf., Izdatel'skii tsentr “Nauka”, Saratov, 2018, 276–279 (in Russian)

[13] Clifford A. H., G. B. Preston, The algebraic theory of semigroups, Amer. Math. Soc., Providence, RI, 1964, 224 pp.

[14] Bogomolov A. M., Salii V. N., Algebraic foundations of the theory of discrete systems, Nauka, M., 1997, 368 pp. (in Russian)

[15] Harary F., Graph Theory, Addison-Wesley, Reading, MA, 1969, 274 pp.

[16] Molchanov V. A., “A universal planar automaton is determined by its semigroup of input symbols”, Semigroup Forum, 82:1 (2011), 1–9 | DOI