We give an algebraic description of Boolean algebras autostable relative to n-decidable presentations. Also, autostable Iλ,μ-algebras are described.
We give an algebraic description of Boolean algebras autostable relative to n-decidable presentations. Also, autostable Iλ,μ-algebras are described.
We provide a complete description of minimal nonzero L-varieties of multiplicative vector spaces over the field ℤ2.
For a partially commutative metabelian group, necessary and sufficient conditions on a defining graph are found under which the intersection of centralizers of two distinct vertices of the graph and the commutator subgroup is trivial.
The spectrum ω(G) of a finite group G is the set of orders of its elements. The following sufficient criterion of nonsolvability is proved: if, among the prime divisors of the order of a group G, there are four different primes such that ω(G) contains all their pairwise products but not a product of any three of these numbers, then G is nonsolvable. Using this result, we show that for q ⩾ 8 and q ≠ 32, the direct square Sz(q) × Sz(q) of the simple exceptional Suzuki group Sz(q) is uniquely characterized by its spectrum in the class of finite groups, while for Sz(32) × Sz(32), there are exactly four finite groups with the same spectrum.
We study A-computable numberings for various natural classes of sets. For an arbitrary oracle A≥T0′, an example of an A-computable family S is constructed in which each A-computable numbering of S has a minimal cover, and at the same time, S does not satisfy the sufficient conditions for the existence of minimal covers specified in [Sib. Math. J., 43, No. 4, 616-622 (2002)]. It is proved that the family of all positive linear preorders has an A-computable numbering iff A′≥T0". We obtain a series of results on minimal A-computable numberings, in particular, Friedberg numberings and positive undecidable numberings.
Let G be a countable saturated model of the theory 𝔗m of divisible m-rigid groups. Fix the splitting G1G2 . . .Gm of a group G into a semidirect product of Abelian groups. With each tuple (n1, . . . , nm) of nonnegative integers we associate an ordinal α = ωm−1nm+ . . . + ωn2 + n1 and denote by G(α) the set ( {G}_1^{n_1}times {G}_2^{n_2}times dots times {G}_m^{n_m} ), which is definable over G in ( {G}^{n_1+dots +{n}_m} ). Then the Morley rank of G(α) with respect to G is equal to α. This implies that RM (G) = ωm−1 + ωm−2 + . . . + 1.
For a wide category K, we introduce the notions of a K-precomplete map and of a K-subspace. Based on these, we create a uniform method for constructing K-completions of T0-spaces.
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