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Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. This paper is the second in a series of papers started in [Algebra and Logic, 61, No. 1, 1-15 (2022)] where we deal with automorphisms of the category of free finitely generated algebras. Here we describe in detail a method of verbal operations. The method provides a characterization of automorphisms of the category of all free finitely generated algebras in a given variety. The characterization plays a crucial role in universal algebraic geometry. We supply the reader with illuminating examples which clarify the method. PubDate: 2022-10-22

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Abstract: Associative rings are considered. By a lattice isomorphism (or projection) of a ring R onto a ring Rφ we mean an isomorphism φ of the subring lattice L(R) of a ring R onto the subring lattice L(Rφ) of a ring Rφ. Let Mn(GF(pk)) be the ring of all square matrices of order n over a finite field GF(pk), where n and k are natural numbers, p is a prime. A finite ring R with identity is called a semilocal (primary) ring if R/RadR ≅ Mn(GF(pk)). It is known that a finite ring R with identity is a semilocal ring iff R ≅ Mn(K) and K is a finite local ring. Here we study lattice isomorphisms of finite semilocal rings. It is proved that if φ is a projection of a ring R = Mn(K), where K is an arbitrary finite local ring, onto a ring Rφ, then Rφ = Mn(K′), in which case K′ is a local ring lattice-isomorphic to the ring K. We thus prove that the class of semilocal rings is lattice definable. PubDate: 2022-10-22

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Abstract: Let ð–ƒ be a class of finite groups which contains a group of order 2 and is closed under subgroups, homomorphic images, and extensions. We define the concept of an ð–ƒ-admissible diagram representing a natural number n. Associated with each n are finitely many such diagrams, and they all can be found easily. Admissible diagrams representing a number n are used to uniquely parametrize conjugacy classes of maximal ð–ƒ-subgroups of odd index in the symmetric group Symn, and we define the structure of such groups. As a consequence, we obtain a complete classification of submaximal ð–ƒ-subgroups of odd index in alternating groups. PubDate: 2022-10-15

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Abstract: We present a new construction of indecomposable type 0 Abelian groups of rank 2. The new construction is used to study degree spectra of such groups. As a corollary, we obtain a new computability-theoretic proof showing that there exist continuum many nonisomorphic type 0 indecomposable Abelian groups of rank 2. PubDate: 2022-10-15

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Abstract: Given a finite undirected graph Γ without loops, we define a sentence Φ(Γ) of group theory. A sequence of graphs Γi is used to obtain a sequence of sentences Φ(Γi). These are employed to determine the Γ-dimension of a group and to study properties of the dimension. Under certain restrictions on a group, the known centralizer dimension is the Γ-dimension for some sequence of graphs. We mostly focus on dimensions defined by using linear graphs and cycles. Dimensions for a number of partially commutative metabelian groups are computed. PubDate: 2022-10-14

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Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Please help us test our new pre-print finding feature by giving the pre-print link a rating. A 5 star rating indicates the linked pre-print has the exact same content as the published article.

Abstract: We study the complexity of index sets with respect to a universal computable numbering of the family of all positive preorders. Let ≤c be computable reducibility on positive preorders. For an arbitrary positive preorder R such that the R-induced equivalence ∼R has infinitely many classes, the following results are obtained. The index set for preorders P with R ≤c P is \( {\sum}_3^0-\mathrm{complete} \) . A preorder R is said to be self-full if the range of any computable function realizing the reduction R ≤c R intersects all ∼Rclasses. If L is a non-self-full positive linear preorder, then the index set of preorders P with P ≡c L is \( {\sum}_3^0-\mathrm{complete} \) . It is proved that the index set of self-full linear preorders is \( {\prod}_3^0-\mathrm{complete} \) . PubDate: 2022-08-06 DOI: 10.1007/s10469-022-09673-z

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Abstract: Let an arbitrary variety of algebras and the category of all free finitely generated algebras in that variety be given. In universal algebraic geometry over an arbitrary variety of algebras, the group of automorphisms of the category of free finitely generated algebras plays an important role. This paper is first in a series where we will deal with the group mentioned. Here we describe properties of automorphisms of the category of all free finitely generated algebras and distinguish two important subgroups: namely, the subgroup of inner automorphisms and the subgroup of strongly stable automorphisms. PubDate: 2022-08-06 DOI: 10.1007/s10469-022-09671-1

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Abstract: The Levi class L(M) generated by the class M of groups is the class of all groups in which the normal closure of every element belongs to M. It is proved that there exists a set of quasivarieties M of cardinality continuum such that \( L\left(\mathrm{M}\right)=L\left(q{H}_{p^s}\right) \) , where \( q{H}_{p^s} \) is the quasivariety generated by the group \( {H}_{p^s} \) , a free group of rank 2 in the variety \( {R}^{p^s} \) of ≤ 2-step nilpotent groups of exponent ps with commutator subgroup of exponent p, p is a prime number, p ≠ 2, s is a natural number, s ≥ 2, and s > 2 for p = 3. PubDate: 2022-08-06 DOI: 10.1007/s10469-022-09674-y

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Abstract: We construct and look at examples of (functional) structures the hereditarily finite superstructures over which have rank of inner constructivizability 0. PubDate: 2022-08-06 DOI: 10.1007/s10469-022-09672-0