By Wieslaw A. Dudek, Valentin S. Trokhimenko

This monograph is the 1st one in English mathematical literature that is dedicated to the speculation of algebras of capabilities of a number of variables. The ebook encompasses a complete survey of major issues of this fascinating thought. specifically the authors learn the proposal of Menger algebras and its generalizations in very systematic means. Readers are supplied with entire bibliography in addition to with systematic proofs of those effects.

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**Example text**

Gn ∈ G. 1, the elements x[g][x ¯ n ] and y[g][y n n ] = y[g][y ¯ ]. So the given inverse for the vector g. ¯ By our assumption, x[g][x ¯ condition is true. Conversely, let the given condition be satisﬁed. If x and y are inverse elements ¯ = gi , gi [y n ][g] ¯ = gi are for a vector (g1 , . . , gn ), then the equalities gi [xn ][g] ¯ n ]. true for all i = 1, . . , n, hence, by the given condition x[g][x ¯ n ] = y[g][y n n ] = x and y[g][y ¯ ] = y, we obtain x = y. 4 gives a known result for semigroups [103].

E) = λg2 (e, . . , e) and, consequently g1 = g2 . So, the mapping P : g → λg is the isomorphism between the algebra (G, o) and the Menger algebra (Λ , O) of full n-place functions, where Λ = {λg | g ∈ G}. 9. Any Menger algebra of rank n is isomorphic to some Menger algebra of n-place functions. Let (G, o) be a Menger algebra of rank n. Let us consider the set Tn (G) of all expressions, called polynomials, in the alphabet G ∪ { [ ], x}, where the square brackets and x do not belong to G, deﬁned as follows: (a) x ∈ Tn (G), (b) if i ∈ {1, .

N ) is the selective semigroup corresponding to (G, o). Since (Gn , ∗) is a group, all its idempotent translations are merely the identity transformations of Gn . Since for all g 1 , . . , g n ∈ Gn there exists a unique g ∈ Gn such that ρi (g i ) = ρ(g), it follows that (g)i = g for all i = 1, . . , n. Therefore the order of Gn is 1. Hence G is a singleton. Further all systems (G, ·, p1 , . . 3 will be called selective semigroups of rank n. The proved theorem gives the possibility to reduce the theory of Menger algebras to the theory of selective semigroups.