ON SOME PROPERTIES OF FE-CLOSURE OPERATOR IN COUNTABLY VALUED LOGIC

Kalinina Inna Sergeevna, Postgraduate student, Moscow State University named after M.V. Lomonosov (1 Leninskie gory street, Moscow, Russia), isenilova@gmail.com

519.716

Background. Functional equations are one of the most widely-used ways to define functions in various areas of mathematics. The article considers the systems of functional equations on multiple functions of countably valued logic and the closure operator, based on existence of a solution for the given systems – FE-closure operator. The work investigates some properties and expressiveness of the FE-closure operator with and without logical connectives.
Materials and methods. The FE-closure operator, considered in the paper, is researched similarly to the previously-known closure operators. Such main notions as closure of set, closed and precomplete class are determinedsimilarly to other closure operators. In proving the author used well-known facts about Boulean functions, permutations in multiple natural numbers, homogeneous function class and conjugation principle for the closure operator.
Results. It is proved that the FE-closure of an empty set with logical connective disjunction coincides with the FE-closure of a ternary discriminator p. It is established that FE-closure of a ternary discriminator p and final constant pool coincides with the set of all functions, self-conjugated relative to any permutations with fixed points coinciding with the given constants. It is shown that the strength of the FEprecomplete class assemblage is at most continuous.
Conclusions. On the basis of the considered properties of the FE-closure operator with and without logical connectives it is possible to estimate its expressiveness. The considered closure operator is a strong closure operator (in comparison, for example, with superposition operators), and nevertheless it causes quite a lot of closed and precomplete classes.

functions of countably valued logic, FE-closure operator.

1. Herbrand J. Sur Compt. rend. Soc. Sci. Lettr. Vars. Classe III. 1931, vol. 24, pp. 12–56.
2. Herbrand J. J. reine und angew. Math. 1931, vol. 166, pp. 1–6.
3. Godel K. Monatsh. Math. und Phys. 1931, vol. 38, pp. 173–198.
4. Marchenkov S. S. Diskretnyy analiz i issledovanie operatsi [Discrete analysis and research of operations]. 2010, no. 4, pp. 18–31.
5. Marchenkov S. S. Vestnik Moskovskogo universiteta. Ser. 15. Vychisl. matem. i kibernet. [Bulletin of Moscow University. Series 15. Calculus mathematics and cybernetics]. 2011, no. 2, pp. 32–39.
6. Marchenkov S. S., Kalinina I. S. Vestnik Moskovskogo universiteta. Ser. 15. Vychislit. matem. i kibernet. [Bulletin of Moscow University. Series 15. Calculus mathematics and cybernetics]. 2013, no. 3, pp. 42–47.
7. Kalinina I. S. Vestnik Moskovskogo universiteta. Ser. 15. Vychislit. matem. i kibernet. [Bulletin of Moscow University. Series 15. Calculus mathematics and cybernetics]. 2014, no. 3, pp. 46–51.
8. Rogers H. Theory of recursive functions and effective computability. Mc Graw-Hill, New York, 1967.
9. Yablonskiy S. V. Vvedenie v diskretnuyu matematiku [Introduction into discrete mathematics]. Moscow: Nauka, 1986.
10. Marchenkov S. S. Diskretnaya matematika [Discrete mathematics]. 2013, no. 4, pp. 13–23.
11. Marchenkov S. S. Problemy kibernetiki [Problems of cybernetics].Issue 39.Moscow:Nauka,1982,pp.85–106.