Почтовый адрес: САФУ, Редакция «Arctic Environmental Research», наб. Северной Двины, 17, г. Архангельск, Россия, 163002
Местонахождение: Редакция «Arctic Environmental Research», наб. Северной Двины, 17, ауд. 1410а, г. Архангельск

Тел: (818-2) 21-61-21 
Сайт: http://aer.narfu.ru/
e-mail: vestnik_est@narfu.ru;
            vestnik@narfu.ru

О журнале

Semigroups Approximation with Respect to Some Ad Hoc Predicates. P. 133–140

Версия для печати

Рубрика: Физика, Математика, Информатика

УДК

512.5

Сведения об авторах

V.V. Dang*, Korabel’shchikova S.Yu.**, Mel’nikov B.F.***
*Vietnam National University, Ho Chi Minh City University of Technology (Ho Chi Minh City, Vietnam)
**Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russian Federation)
***Center of Information Technologies and Systems for Executive Power Authorities (Moscow, Russian Federation)
address: Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation;
e-mail: kmv@atnet.ru

Аннотация

The problem of semigroups approximation with respect to various predicates has been investigated by many scientists. Some necessary and sufficient conditions for the semigroups approximation with respect to such predicates as “equality”, “membership of an element to a subsemigroup”, “regular conjugation relation”, “Green ratio of L-, R-, H- and D-equivalence”, “membership of an element to a monogenic subsemigroup”, etc. were obtained. However, there were practically no results on the conditions of approximation with respect to the predicate of membership of an element to a subgroup of a given semigroup. The paper presents the necessary and sufficient condition for approximation with respect to this predicate. We constructed a special semigroup acting the role of a minimal approximation semigroup for many predicates. This semigroup has neither identity nor additive identity. It contains an infinite number of idempotents, and the presence of each idempotent is mandatory. By this semigroup, we have successfully solved the problem of approximation with respect to the predicate of membership of an element to a subsemigroup. A class of semigroups is also described, for which it is the minimal approximation semigroup. We obtained a criterion for the approximation of a semigroup with respect to the Green H-equivalence. The problem of algebraic systems approximating with respect to a predicate consists of three components: a set of algebraic systems (groups, semigroups, etc.); set of predicates; set of functions (homomorphisms, continuous mappings, etc.). The change of one of these components determines a new line of research.

Ключевые слова

semigroups approximation, approximation with respect to the predicate, minimal semigroup of approximation
Скачать статью (pdf, 1.9MB )

Список литературы

  1. Mal’tsev A.I. Izbrannye trudy. T. 1. Klassicheskaya algebra [Selectas. Vol. 1. Classical Algebra]. Moscow, 1976. 484 p. 
  2. Lesokhin M.M. Ob approksimatsii polugrupp otnositel’no predikatov [On the Approximation of Semigroups with Respect to Predicates]. Uchenye zapiski LGPI im. A.I. Gertsena [Ttransactions of the Leningrad State Pedagogical Institute], 1971, vol. 404, pp. 191–219. 
  3. Golubov E.A. O finitnoy approksimiruemosti otdelimykh estestvenno lineyno uporyadochennykh kommutativnykh polugrupp [On Finite Approximability of Separable Naturally Linearly Ordered Commutative Semigroups]. Izvestiya vysshikh uchebnykh zavedenii. Matematika, 1969, no. 2, pp. 23–31. 
  4. Mamikonyan S.G. Mnogoobraziya finitno-approksimiruemykh polugrupp [Varieties of Finitely Approximable Semigroups]. Matematicheskii Sbornik, 1972, vol. 88(130), no. 3(7), pp. 353–359. 
  5. Lesokhin M.M., Golubov E.A. O finitnoy approksimiruemosti kommutativnykh polugrupp [On the Finite Approximability of Commutative Semigroups]. Matematicheskie zapiski Ural’skogo universiteta, 1966, vol. 5, no. 3, pp. 82–90. 
  6. Kublanovskiy S.I. O finitnoy approksimiruemosti predmnogoobraziy polugrupp otnositel’no predikatov [On the Finite Approximability of the Semigroups Pre-Varieties with Respect to Predicates]. Sovremennaya algebra. Gruppoidy i ikh gomomorfizmy [Abstract Algebra. Groupoids and Their Homomorphisms]. Leningrad, 1980, pp. 58–88. 
  7. Ignat’eva I.V. SH-approksimatsiya polugrupp konechnymi kharakterami [SH-Approximation of Semigroups by Finite Characters]. Sovremennaya algebra: mezhvuzovskiy sbornik nauchnykh trudov. Vyp. 1 [Abstract Algebra: Inter- University Collection of Scientific Papers. Iss. 1]. Rostov-on-Don, 1996, pp. 25–30. 
  8. Tutygin A.G., Yashina E.Yu. Zavisimost’ usloviy approksimatsii polugrupp po nekotorym predikatam [Dependence of Conditions for Semigroups Approximation with Respect to Certain Predicates]. Sovremennaya algebra: mezhvuzovskiy sbornik nauchnykh trudov. Vyp. 3 [Abstract Algebra: Inter-University Collection of Scientific Papers. Iss. 3]. Rostov-on-Don, 1998, pp. 136–141. 
  9. Dang V.V., Korabel’shchikova S.Yu., Mel’nikov B.F. O zadache nakhozhdeniya minimal’noy polugruppy approksimatsii [On the Problem of Finding Minimum Semigroup of Approximation]. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki [University Proceedings. Volga Region. Physical and Mathematical Sciences], 2015, no. 3(35), pp. 88–98. 
  10. Zyablitseva L.V., Korabel’shchikova S.Yu., Popov I.N. Nekotorye spetsial’nye polugruppy i ikh gomomorfizmy [Some Special Semigroups and Their Homomorphisms]. Arkhangelsk, 2013. 128 p. 
  11. Petrich M. Introduction to Semigroups. USA, Columbus, Ohio, 1973. 193 p. 
  12. Lyapin E.S. Polugruppy [Semigroups]. Moscow, 1960. 592 p. 
  13. Clifford A.H., Preston G.B. The Algebraic Theory of Semigroups. USA, Providence, 1972. 225 p. 
  14. Dang V.V. Problema minimalizatsii polugruppy approksimatsii i SH-approksimatsii [The Problem of Minimization of Approximation Semigroup and SH-Approximation]. Sovremennaya algebra: mezhvuzovskiy sbornik nauchnykh trudov. Vyp. 3 [Abstract Algebra: Inter-University Collection of Scientific Papers. Iss. 3]. Rostov-on-Don, 1999, pp. 43–48.