МАТЕМАТИКА УДК 512.5 DOI: 10.17238/issn2541-8416.2017.17.2.133 SEMIGROUPS APPROXIMATION WITH RESPECT TO SOME AD HOC PREDICATES1 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) 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. <...> 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. <...> 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.). <...> Keywords: semigroups approximation, approximation with respect to the predicate, minimal semigroup of approximation. 1This research is funded by Ho Chi Minh City University of Technology under grant number T-KHUD-2016-109. <...> Corresponding author: Svetlana Korabel’shchikova, address: Naberezhnaya Severnoy Dviny, 17, Arkhangelsk, 163002, Russian Federation; e-mail: kmv@atnet.ru For citation: Dang V.V., Korabel'shchikova S.Yu., Mel'nikov B.F. Semigroups Approximation with Respect to Some Ad Hoc Predicates. <...> С. 133–140 A common concept of approximation of the algebraic system is represented by the Russian academician A.I. Mal’tsev [1, pp. 450–462]. <...> In the article, Mal’tsev demonstrates a connection between a finite approximation of the algebraic system with respect to a given predicate and a problem of its solvability in the system. <...> A notion of a finitely approximable semigroup is also mentioned with some results on the semigroup approximation. <...> The problem of semigroup approximation <...>