If M is a set of finite groups, then a group G is said to be saturated with the set M (saturated with groups in M) if every finite subgroup of G is contained in a subgroup isomorphic to some element of M. It is proved that a periodic group with locally finite centralizers of involutions, which is saturated with a set consisting of groups L4(q), where q is odd, is isomorphic to L4(F) for a suitable field F of odd characteristic.
New constructions of prime binary Lie superalgebras are presented. Based on one of these, we construct the first example of a prime binary Lie algebra that is not a Mal’tsev algebra.
It is proved that the associated Lie algebra of the Mal’tsev ℚ-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. It is also shown that the homology of this completed Lie algebra is of uncountable dimension in each degree.
This paper is a continuation of [Algebra and Logic, 58, No. 6, 447-469 (2019)] where we constructed polynomial-time presentations for the field of complex algebraic numbers and for the ordered field of real algebraic numbers. Here we discuss other known natural presentations of such structures. It is shown that all these presentations are equivalent to each other and prove a theorem which explains why this is so. While analyzing the presentations mentioned, we introduce the notion of a quotient structure. It is shown that the question whether a polynomial-time computable quotient structure is equivalent to an ordinary one is almost equivalent to the P = NP problem. Conditions are found under which the answer is positive.
We prove a theorem stating the following. Let G be a periodic group saturated with finite Frobenius groups with complements of even order, and let i be an involution of G. If, for some elements a, b ∈ G with the condition |a| · |b| > 4, all subgroups 〈a, bg〉, where g ∈ G, are finite, then G = A λ CG(i) is a Frobenius group with Abelian kernel A and complement CG(i) whose elementary Abelian subgroups are all cyclic.
It is proved that every two predicate calculi of finite rich signatures are algebraically virtually isomorphic, i.e., some of their Cartesian extensions are algebraically isomorphic. As an important application, it is stated that for predicate calculi in any two finite rich signatures, there exists a computable isomorphism between their Tarski–Lindenbaum algebras which preserves all model-theoretic properties of algebraic type corresponding to the real practice of research in model theory.
Presented by the Dissertation Council D 003.015.02
We look at the complexity of the existence problem for a Horn sentence (identity, quasi-identity, ∀-sentence, ∃-sentence) equivalent to a given one. It is proved that if the signature contains at least one symbol of arity k ≥ 2, then each of the problems mentioned is an m-complete ( {Sigma}_1^0 ) set.
We consider the lattices of extensions of three logics: (1) modal bilattice logic; (2) full Belnap–Dunn bimodal logic; (3) classical bimodal logic. It is proved that these lattices are isomorphic to each other. Furthermore, the isomorphisms constructed preserve various nice properties, such as tabularity, pretabularity, decidability or Craig’s interpolation property.
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