Abstract. The equational theories were studied in many works (see [4], [5], [6], [7]). Let r be a type of Abelian groups. In this paper we consider the extentions of the equational theory Ex(Q n ) defined by so called externally compatible identities of Abelian groups and the identity x n « y n . The equational base of this theory was found in [3]. We prove that each equational theory Cn(Ex(Q n ) U {φ « Φ}), where φ « φ is an identity of type r, is equal to the extension of the equational theory Cn(Ex(Q n ) U E), where E is a finite set of one variable identities of type r.The notation in this paper are the same as in [1]. PreliminariesLet r : {·, -1 }->N be a type of Abelian groups where τ(·) = 2, r( _1 ) = l. By Q n we denote the class of all Abelian groups satisfying the identityThe identity of type τ is externally compatible (see [2]) if it is one of the form χ « χ or of the form φχ · >2 w φι · Φ2, Φΐ is the equational theory. Let Id(r) be a set of all identities of type r. By Cn(E), where Σ Ç Id(r), we denote the deductive closure of Σ.It is well known fact, that the lattice of all equational theories extendingis dually isomorphic to the lattice of all natural divisors of η with divisibility relation. It implies that Cn{Id{G n ) The algorithm presented above neglects the structure of identities, and that is why it is useless in the case of extensions of the theory Ex(Q n ).Using the Galois connection between algebras and identities we have that the lattice of all equational theories of type τ is dually isomorphicto the lattice of all varieties of the same type. So, if we know all theories, where φ and ψ are terms of type r, we can describe all subvarieties of the variety defined by all externally compatible identities of the variety Q n . The extension of the theory Ex(Q n )In this paper, as in [3], by x° we denote χ · χ -1 . Let us consider the following identities: Sets of identities satisfied in Abelian groups 449Let us consider the identity (1). The following lemma is obvious. Proof. Without losing generality we can assume that j = 1. Let S\ = Cn(Ex(G n )U{ (2) ,k s ).Putting Xj = x\' for j G {2,..., s} in the identity (1) we get, that (xi « Xi · g Si and thus (xi · x? « Xl · x? · € S u so we have (χχ « ιξ · xf +1 ) G Si. Finally, we have S 2 Ç Si. To prove the opposite inclusion let us note, that from the condition (χι « x? · xf +1 ) G S 2 it follows that (x° ~ x? · xf) G S 2 . The immediate consequence of these conditions is (χι « χξ • χχ) G S 2 . The definition of d implying that for each j from the set {2,..., s} a number d is a divisor of kj. Hence there exist elementsp 2 ,...,p s in the set Z n such that kj = pj • d. As a result of the condition (x° « x° · xf ) G S 2 we have that for each je {2,..., s} the identity x° « x° · x^y d belongs to S 2 . From the fact that where d = (ki,..., kj-1, kj+i,..., ks). (fci,..., kj-1, kj -1, kj+i,..., ks).Proof. The proof of this lemma is analogously to the proof of the Lemma 2.• Let we study the identity (4).
The lattices of varieties were studied in many works (see [4], [5], [11], [24], [31]). In this paper we describe the lattice of all subvarieties of the variety G n Ex defined by so called externally compatible identities of Abelian groups and the identity x n ≈ y n .The notation in this paper is the same as in [2].
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