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By Waclaw Sierpinski, I. N. Sneddon, M. Stark

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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.