Vol 55, No 4 (2016)

It is proved that every computable ordinal has a degree of autostability. We construct new examples of degrees of autostability in the class of linear orders and in the class of linearly ordered Abelian groups.

Computable presentations for projective planes are studied. We prove that the problem of computable categoricity is ∏₁¹-complete for the following classes of projective planes: Pappian projective planes, Desarguesian projective planes, arbitrary projective planes.

We introduce a classification of extensions of Johansson’s minimal logic J that extends the classification of superintuitionistic logics proposed by T. Hosoi. It is proved that the layer number of any finitely axiomatizable logic is effectively computable. Every layer over J has a least logic. It is stated that each layer has finitely many maximal logics, and minimal and maximal logics of all layers are recognizable over J.

Let a group G have an Abelian normal subgroup A; put $\bar{G}$ = G/A and $\bar{g}$ = gA for g ∈ G. We can think of A as a right ℤ$\bar{G}$-module and define the action of an element $u={\alpha }_1{\bar{g}}_1$ +…+ ${\alpha }_n{\bar{g}}_n$ ∈ ℤ$\bar{G}$ on a ∈ A by a formula a^u = ${\left(a^{g_1}\right)}^{{\alpha }_1}$ ·…· ${\left(a^{g_n}\right)}^{{\alpha }_n}$ ; here $a^{g_i}=g_i^{-1}a{g}_i$. Denote by ${\mathrm{\Theta }}_{\mathbb{Z}\bar{G}}$(A) the annihilator of A in the ring ℤ$\bar{G}$, which is a two-sided ideal. Let $R=\mathbb{Z}\bar{G}/{\mathrm{\Theta }}_{\mathbb{Z}\bar{G}}(A)$. A subgroup A can also be treated as an R-module. We give a criterion for the existence of an R-decomposition of G over A, i.e., the possibility of embedding G in a semidirect product $\bar{G}$·D, where D is an R-module. It is also proved that an R-decomposition always exists in one important case.

We prove some results in universal algebraic geometry over algebraic structures of arbitrary functional languages with relation ≠ adjoined.