Algebra and Logic最新文献

We study enriched Boolean algebras. We describe subalgebras of Boolean algebras determined by fixed elements of automorphisms. We construct a continuum of superatomic Boolean algebras with a distinguished subalgebra whose elementary theories have no prime model. We investigate semigroups of elementary types of Boolean algebras with a distinguished subalgebra and of Boolean algebras with distinguished ideals.

Let G be the group of all bounded permutations of the set of natural numbers ({mathbb{N}}). Suppose that a group H contains elements h, h₁, . . . , h_n, . . . such that |h| = ∞, ({h}_{1}^{p}) = h, ({h}_{2}^{p}) = h₁, . . . , ({h}_{n}^{p}) = h_n−1, . . . , where p is a prime number. We prove that H is not isomorphic to a subgroup of G.

We prove that G₂(2^m) and ³D₄(2^3m) are automorphism groups of regular polytopes of rank n = 4, 5, except for the group ³D₄(2^3m) when n = 5 and m is a power of 3. The minimal number of generating conjugate involutions whose product equals 1 is found, and the existence of new regular polytopes of rank 3 with these automorphism groups is established.

We consider the problem of existence of single-valued, computable, positive, and negative representations of the family of all computably enumerable subsets on admissible structures. Additionally, a description of the properties of the admissible structures under consideration is given.

We consider intersections of π-Hall subgroups in a finite D_π-group all of whose composition factors are sporadic or alternating groups. The main result of the paper in the case under consideration confirms Conjecture 18.31 from the Kourovka Notebook, and also gives an answer to Question 20.123(b).

We describe relational structures of finite signature representable by means of infinite time Blum–Shub–Smale machines in terms of Borel hierarchy on Baire and Cantor Polish spaces.

A strongly constructivizable model S is decidably categorical if for any decidable copies A and B of S, there exists a computable isomorphism from A onto B. The paper investigates the complexity of families of decidably categorical models belonging to familiar finitely axiomatizable classes. For a non-zero natural number n, the index set of decidably categorical linear orders elementarily equivalent to the ordinal ωⁿ is an m-complete ({Sigma}_{2text{n}+2}^{0})-set. The index set of decidably categorical structures which are elementarily equivalent to ωⁿ, enriched by the successor relation, is an m-complete ({Sigma}_{2text{n}+1}^{0})-set.

An embedding of the hereditarily finite superstructure over the ordered field of real numbers into the set of reals is constructed which takes Σ-subsets to sets computable by infinite time Blum-Shub-Smale machines (ITBMs). A notion of ITBM-constructivizable structure is introduced. It is proved that constructivizability of an arbitrary algebraic structure over the ordered field of real numbers implies its ITBM-constructivizability. We obtain a theorem on the existence of ITBM-constructivizable models of the cardinality of the continuum for countable consistent theories with infinite models.

We investigate the existence of hybrids of Jonsson theories in various first-order languages. A general framework for hybrids is constructed, defined by ∀∃-sentences true in the Cartesian product of semantic models. For theories in the same language, examples of hybrids are provided; necessary and sufficient conditions for the existence of a hybrid in a functional signature are established; it is shown that a hybrid can inherit properties of the original theories (perfection, categoricity, stability). For theories in different languages, the existence of hybrids is proved for arbitrary enrichments of semantic models to the joint language, and an amalgam is constructed. It is shown that hybrids can differ depending on the enrichments.

We study the following conjecture, which is a sharp analogue of the well-known Baer–Suzuki theorem for the π-radical of a finite group. For an arbitrary set π of primes not containing all primes, let r be the smallest prime not in π. Set m = r if r ⩽ 3 and m = r − 1 if r > 3. Then, in a finite group G, the largest normal π-subgroup always coincides with the set of elements x such that any m conjugates of x generate a π-subgroup. To date, this conjecture has been confirmed for any finite group whose every nonabelian composition factor is isomorphic to a sporadic, alternating, linear, or unitary simple group, or to one of the groups ²B₂(q), ²G₂(q), ²F₄(q)′, G₂(q), or ³D₄(q). It is proved that the simple symplectic groups S_2n(q) can be added to this list.