Algebra and Logic最新文献

On a Cartan group ({mathbb{K}}) equipped with a Carnot–Carathéodory metric d_cc, we find the exact value of a constant in the (1, q₂)-generalized triangle inequality for its Box-quasimetric. It is proved that any two points x, y ∈ ({mathbb{K}}) can be joined by a horizontal k-broken line ({L}_{x,y}^{k}), k ≤ 6; moreover, the length of such a broken line ({L}_{x,y}^{k}) does not exceed the quantity Cd_cc(x, y) for some constant C not depending on the choice of x, y ∈ ({mathbb{K}}). The value 6 here is nearly optimal.

A finitely generated group G, which acts on a tree so that all edge stabilizers are infinite cyclic groups and all vertex stabilizers are free rank 2 Abelian groups, is called a tubular group. Every tubular group is isomorphic to the fundamental group π₁(𝒢) of a suitable finite graph 𝒢 of groups. We prove a criterion for residual π-finiteness of tubular groups presented by trees of groups. Also we state a criterion for residual p-finiteness of tubular groups whose corresponding graph contains one edge outside a maximal subtree.

A Levi class (Lleft(mathcal{M}right)) generated by a class (left(mathcal{M}right)) of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to (left(mathcal{M}right)). Let p be a prime and p ≠ 2, let H_p be a free group of rank 2 in the variety of nilpotent groups of class at most 2 with commutator subgroup of exponent p, and let qH_p be the quasivariety generated by the group H_p. It is shown that there exists a set of quasivarieties (mathcal{M}) of cardinality continuum such that (Lleft(mathcal{M}right)) = L(qH_p). Let s be a natural number, s ≥ 2. We specify a system of quasi-identities defining L(q(H_p, ({Z}_{{p}^{s}}))), and prove that there exists a set of quasivarieties (mathcal{M}) of cardinality continuum such that (Lleft(mathcal{M}right)) = L(q(H_p, ({Z}_{{p}^{s}}))), where ({Z}_{{p}^{s}}) is a cyclic group of order p^s; q(H_p, ({Z}_{{p}^{s}})) is the quasivariety generated by the groups H_p and ({Z}_{{p}^{s}}.)

We find a bound for the length of a conjunction of some propositional formulas, for which every unsatisfiable formula contains an unsatisfiable subformula. In particular, this technique applies to formulas in conjunctive normal form with restrictions on the number of true literals within every elementary disjunction, as well as for 2-CNFs, for symmetric 3-CNFs, and for conjunctions of voting functions in three literals. A lower bound on the rank of some matrices is used in proofs.

Relations between some constructions based on incidence rings and group rings are considered.

If a certain condition holds for a quasivariety K then K contains continuum many subquasivarieties having a finitely partitionable ω-independent quasi-equational basis relative to K. This is true, in particular, for each almost ff-universal quasivariety K.

It is proved that, for any natural n, the subalgebra generated by words of length divisible by n on generators (the Veronese n-subalgebra) in a free finitely generated alternative algebra is finitely generated.

It is proved that if G is a group without elements of order 2, and the normal closure of every 2-generated subgroup of G is a nilpotent group of class at most 3, then G will be a nilpotent group of class at most 4. It is also shown that the restriction on second-order elements cannot be lifted.