Algebra and Logic最新文献

The paper is devoted to the construction of a finitely presented infinite nilsemigroup satisfying the identity x⁹ = 0. We describe an algorithm reducing arbitrary semigroup words to a canonical form. We prove that any word containing a subword of period 9 can be reduced to zero using the defining relations. At the same time, there exist words corresponding to arbitrarily long paths whose length does not decrease, demonstrating that the constructed semigroup is infinite.

Extending a result of Zacharov, we show that every nonzero enumeration degree consists of infinitely many s-degrees. In fact, we show that there is no minimal s-degree inside any nonzero enumeration degree. This answers open questions in the literature raised by Cooper and Batyrshin.

We study partial mappings on natural numbers, the graphs of which are coenumerable. Such mappings are referred to as negative mappings. We show that any 0′-computable partial function is represented as the superposition of two negative ones. We also show that the inverse semigroup of all 0′-computable partial injective mappings is generated by its negative elements; moreover, any its element is equal to the product of its two negative elements. We show that the group of all 0′-computable permutations is generated by its negative elements. We obtain sufficient conditions for the representability of 0′- computable permutations in the form of the superposition of two negative permutations.

We calculate the second cohomology group for the four-dimensional simple Jordan superalgebra 𝒥osp_1|2(𝔽) by proving that its second cohomology group with coefficients in the regular representation is isomorphic to ({mathbb{F}}dot{+}0). We show (without proof) that for the four-dimensional simple Jordan superalgebra 𝒟_t with t ≠ 0, the superalgebra (V, f) of a superform with n = 1 and m = 1, and ℳ_1|1(𝔽)⁽⁺⁾ the respective second cohomology groups (with coefficients in the regular representation) are isomorphic to ({mathbb{F}}dot{+}0).

We give a description of 2-decidable Boolean algebras with one distinguished ideal in terms of the computability of some set of predicates on a given algebra.

We construct an Anick type wild automorphism in a 3-generated free Poisson algebra which induces a tame automorphism in a 3-generated polynomial algebra. We also show that this automorphism is stably tame.

It is proved that, for any vector space V over a field f of finite dimension at least 3, the projective space P(V) (the set of all subspaces of V equpped with a binary predicate of inclusion) is regularly injectively bi-interpretable with the field F.

For the solvable Baumslag–Solitar group BS(1, n) (n > 1), we define the divisible completion BSd(1, n). We describe groups that are elementarily equivalent to the group BSd(1, n), find the axiomatics of the theory of the group BSd(1, n), and prove the decidability of this theory.

We study the famous Mal’tsev correspondence between nilpotent k-groups G and nilpotent Lie k-algebras L over a field k of characteristic zero from the model-theoretic, algebro-geometric, and algorithmic viewpoints. It is proved that, in this case, a group G and the corresponding Lie algebra L(G) are bi-interpretable by equations in each other. This gives a much more precise description of the correspondence, which implies that, in addition to the classical categorical properties, the group G and the algebra L(G) share many more algebraic, algorithmic, and model-theoretic properties.

Answering a question of A. V. Vasil’ev, we show that each finite symmetric (or alternating) group H is a retract of any group containing H as a verbally closed subgroup.