By Wieslaw A. Dudek, Valentin S. Trokhimenko

This monograph is the 1st one in English mathematical literature that's dedicated to the speculation of algebras of features of a number of variables. The publication features a finished survey of major themes of this fascinating concept. particularly the authors examine the inspiration of Menger algebras and its generalizations in very systematic method. Readers are supplied with entire bibliography in addition to with systematic proofs of those effects.

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**Example text**

6) holds for every idempotent e ∈ G and every idempotent vector a¯ ∈ Gn , (c) the idempotents of the algebra (G, o) are diagonal-commutative, and idempotent vectors of (G, o) are pseudo-commutative. Proof. Suppose that (G, o) is a v-regular Menger algebra of rank n and that condition (a) is true. , the Menger algebra (G, o) is inverse. It is known that idempotents are diagonal-commutative in any inverse Menger algebra. 6) is true. This shows that (a) implies (b). Now let (b) hold. Clearly, idempotents of (G, o) are diagonal-commutative, therefore we have only to show that idempotent vectors are pseudo-commutative.

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.

1 33 Deﬁnitions and fundamental notions Proof. We prove only (b) because the proof of (a) is analogous. Let g1 ≡ g2 (εm ρ, A ) for some m. If g1 ≡ g2 (ε ρ, A ) and g1 , g2 ∈ A , then (u[w¯ |i g1 ], u[w¯ |i g2 ]) ∈ A × A ⊂ ε ρ, A . As A is an l-ideal, then suppose that (g1 , g2 ) ∈ εm ρ, A for some m. 11) we have u[w¯ |i g2 ] ∈ A and vice versa. Thus, the elements u[w¯ |i g1 ], u[w¯ |i g2 ] belong or do not belong to A simultaneously. If these elements belong to A, then (g1 , g2 ) ∈ εm ρ, A ◦εm ρ, A .