Vol 56, No 5 (2017)

We study a representation of the virtual braid group VB_n into the automorphism group of a free product of a free group and a free Abelian group, proposed by S. Kamada. It is proved that the given representation is equivalent to the representation constructed in [http://arxiv.org/abs/1603.01425]; i.e. the kernels of these representations coincide.

Extensions of Johansson’s minimal logic J are considered. It is proved that families of negative and nontrivial logics and a series of other families are strongly decidable over J. This means that, given any finite list Rul of axiom schemes and rules of inference, we can effectively verify whether the logic with axioms and schemes, J + Rul, belongs to a given family. Strong recognizability over J is proved for known logics Neg, Gl, and KC as well as for logics LC and NC and all their extensions.

A group G is said to be rigid if it contains a normal series G = G₁ > G₂ > … > Gm > G_m+1 = 1, whose quotients G_i/G_i+1 are Abelian and, treated as right ℤ[G/G_i]-modules, are torsion-free. A rigid group G is divisible if elements of the quotient G_i/G_i+1 are divisible by nonzero elements of the ring ℤ[G/G_i]. Every rigid group is embedded in a divisible one. We prove two theorems. Theorem 1 says that the following three conditions for a group G are equivalent: G is algebraically closed in the class Σ_m of all m-rigid groups; G is existentially closed in the class Σ_m; G is a divisible m-rigid group. Theorem 2 states that the elementary theory of a class of divisible m-rigid groups is complete.