Sh-Weak Duality of Semigroups and Minimum Semi-Group of Sh-Approximation

About the class of approximation by generalized characters

“…Definition 2.3. 1,15 Let K 1 and K 2 be two classes of algebraic structures with the same type and P be a predicate. The class K 2 is said to be a class of approximation with respect to the predicate P for the class K 1 , if every algebraic structure from K 1 is approximable into K 2 with respect to P. Definition 2.4.…”

“…The class K 2 is said to be a class of approximation with respect to the predicate P for the class K 1 , if every algebraic structure from K 1 is approximable into K 2 with respect to P. Definition 2.4. 1,15 If every structure from the class K 1 can be embedded into some structure from a class K 2 , then we write K 1 ≺ K 2 It's obvious that, if K 1 is a class of approximation for the class K and K 1 ≺ K 2 , then K 2 is also a class of approximation for the class K. Definition 2.5. 1,15 Let S and K be two classes of algebraic structures with the same type and P be a predicate.…”

“…1,15 If every structure from the class K 1 can be embedded into some structure from a class K 2 , then we write K 1 ≺ K 2 It's obvious that, if K 1 is a class of approximation for the class K and K 1 ≺ K 2 , then K 2 is also a class of approximation for the class K. Definition 2.5. 1,15 Let S and K be two classes of algebraic structures with the same type and P be a predicate. The class K is called a minimal class of approximation for the class S with respect to P, if the following two conditions hold: (i) The class K is a class of approximation for the class S;…”

“…A problem of approximation of algebraic structures with respect to various predicates has been proposed and studied by Professor M. M. Lesokhin 6,7 and his students [1][2][3][4][5][6][7][8][9][10][11][12][13][14] . They have found a necessary and sufficient conditions for approximation of algebraic structures with respect to many important predicates, such as "equality of two elements", "belonging of an element in a substructure", "belonging of an element in the group", "the relation of regular conjugation", "Green l−, R−, h−, d− equivalency", "belonging of an element in the monogenic subsemigroup", etc.…”

“…Some theorems related to the problem of constructing an approximation class are obtained. The problem in question is much more complicated and actual than the approximation problem we have been studying before [1][2][3][4][5] . The results of the description of the approximation class play an important role in studying the solvability problem of the predicate P in the class of semigroups K. In particular, if the approximation class consists of finite semigroups, then this problem is solved positively.…”