By Wieslaw A. Dudek, Valentin S. Trokhimenko

This monograph is the 1st one in English mathematical literature that's dedicated to the idea of algebras of capabilities of a number of variables. The e-book includes a entire survey of major issues of this attention-grabbing idea. particularly the authors research the thought of Menger algebras and its generalizations in very systematic approach. Readers are supplied with whole bibliography in addition to with systematic proofs of those effects.

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Xn ) = x0 · n x1 x2 · · · xn is a Menger algebra for which its diagonal semigroup is a commutative group. 1. If a Menger algebra (G, o) has a right diagonal unit e, then every element c ∈ G satisfying the identity e = e[cn ] is also a right diagonal unit. Proof. Indeed, for every b ∈ G we have b = b[en ] = b[e[cn ] · · · e[cn ]] = b[en ][cn ] = b[cn ], which means that also c is a right diagonal unit. It is not diﬃcult to see that the following two lemmas are true. 2. , b[xn ] = b[y n ] −→ x = y for all b, x, y ∈ G.

For this, consider two (n+1)-ary operations o1 (x0 , x1 , . . , xn ) = x0 +x1 and o2 (x0 , x1 , . . , xn ) = x0 + x2 deﬁned on a nontrivial commutative semigroup (G, +). Then, as it is easy to see, (G, o1 ) and (G, o2 ) are an example of non-isomorphic Menger algebras of rank n with the same diagonal semigroup (G, +). 6. A Menger algebra (G, o) of the rank n is called unitary, if there exist elements e1 , . . , en ∈ G, called selectors, such that x[e1 · · · en ] = x and ei [x1 · · · xn ] = xi for all x, x1 , .

4, this semigroup is a diagonal semigroup of a Menger algebra with the operation x[y1 · · · yn ] = x·f (y1 , . . , yn ). A left unit e of this semigroup is a left diagonal unit of the Menger algebra. Indeed, for any x ∈ G we have e[xn ] = e · f (x, . . , x) = f (x, . . , x) = x, which completes the proof. 24 Chapter 2 Menger algebras of functions Note that non-isomorphic Menger algebras may have the same diagonal semigroup. For this, consider two (n+1)-ary operations o1 (x0 , x1 , . . , xn ) = x0 +x1 and o2 (x0 , x1 , .