Vol 54, No 6 (2016)

It is proved that every nontrivial linearly minimal Jordan algebra of characteristic other than 2 is a division algebra. Then Zel’manov’s classification of Jordan division algebras is applied to show that such an algebra is a field.

Let V_∞be a standard computable infinite-dimensional vector space over the field of rationals. The lattice\( \mathfrak{L} \)(V_∞) of computably enumerable vector subspaces of V_∞and its quotient lattice modulo finite dimension,\( \mathfrak{L} \)^*(V_∞), have been studied extensively. At the same time, many important questions still remain open. In 1998, R. Downey and J. Remmel posed the question of finding meaningful orbits in\( \mathfrak{L} \)^*(V_∞) [4, Question 5.8]. This question is important and difficult and its answer depends on significant progress in the structure theory for the lattice\( \mathfrak{L} \)^*(V_∞), and also on a better understanding of its automorphisms. Here we give a necessary and sufficient condition for quasimaximal (hence maximal) vector spaces with extendable bases to be in the same orbit of\( \mathfrak{L} \)^*(V_∞). More specifically, we consider two vector spaces, V₁and V₂, which are spanned by two quasimaximal subsets of, possibly different, computable bases of V_∞. We give a necessary and sufficient condition for the principal filters determined by V₁and V₂in\( \mathfrak{L} \)^*(V_∞) to be isomorphic. We also specify a necessary and sufficient condition for the existence of an automorphism Φ of\( \mathfrak{L} \)^*(V_∞) such that Φ maps the equivalence class of V₁to the equivalence class of V₂. Our results are expressed using m-degrees of relevant sets of vectors. This study parallels the study of orbits of quasimaximal sets in the lattice ε of computably enumerable sets, as well as in its quotient lattice modulo finite sets, ε^*, carried out by R. Soare in [13]. However, our conclusions and proof machinery are quite different from Soare’s. In particular, we establish that the structure of the principal filter determined by a quasimaximal vector space in\( \mathfrak{L} \)^*(V_∞) is generally much more complicated than the one of a principal filter determined by a quasimaximal set in ε^*. We also state that, unlike in ε^*, having isomorphic principal filters in\( \mathfrak{L} \)^*(V_∞) is merely a necessary condition for the equivalence classes of two quasimaximal vector spaces to be in the same orbit of\( \mathfrak{L} \)^*(V_∞).

The notion of Turing computable embedding is a computable analog of Borel embedding. It provides a way to compare classes of countable structures, effectively reducing the classification problem for one class to that for the other. Most of the known results on nonexistence of Turing computable embeddings reflect differences in the complexity of the sentences needed to distinguish among nonisomorphic members of the two classes. Here we consider structures obtained as sums. It is shown that the n-fold sums of members of certain classes lie strictly below the (n+1)-fold sums. The differences reflect model-theoretic considerations related to Morley degree, not differences in the complexity of the sentences that describe the structures. We consider three different kinds of sum structures: cardinal sums, in which the components are named by predicates; equivalence sums, in which the components are equivalence classes under an equivalence relation; and direct sums of certain groups.

Presented by the Program Committee of the Conference “Mal’tsev Readings”