We construct and look at examples of (functional) structures the hereditarily finite superstructures over which have rank of inner constructivizability 0.
We construct and look at examples of (functional) structures the hereditarily finite superstructures over which have rank of inner constructivizability 0.
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} ).
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 ( Lleft(mathrm{M}right)=Lleft(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.
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.
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