In this paper are described the groups of automorphisms of semigroup End(W (X)), where W (X) is free commutative or free associative algebra.H. Then arbitrary congruence T in W determines the set of points in affine space Hom(W, H):

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Let [Formula: see text] be the variety of associative algebras over a field K and A = K 〈x1,…, xn〉 be a free associative algebra in the variety [Formula: see text] freely generated by a set X = {x1,…, xn}, End A the semigroup of endomorphisms of A, and Aut End A the group of automorphisms of the semigroup End A. We prove that the group Aut End A is generated by semi-inner and mirror automorphisms of End A. A similar result is obtained for the automorphism group Aut [Formula: see text], where [Formula: see text] is the subcategory of finitely generated free algebras of the variety [Formula: see text]. The later result solves Problem 3.9 formulated in [17].

In this paper, we study the geometric equivalence of algebras in several varieties of algebras. We solve some of the problems formulated in [2], in particular, that of geometric equivalence for real-closed fields and finitely generated commutative groups.