arXiv subjects
Publications and source records attributed to S. F. Soprunov.
We investigate the definability (reducts) lattice of the order of integers and describe a sublattice generated by relations 'between', 'cycle', 'separation', 'neighbor', '1-codirection', 'order' and equality'. Some open questions are proposed.
We consider a structure $\mathcal {M} = \langle \mathbb N, \{Tr,<\} \rangle$, where the relation $Tr(a,x,y)$ with a parameter $ a$ defines a family of trees on $\mathbb N$ and $<$ is the usual order on $\mathbb N$. We show that if the elementary theory of $\mathcal M$ is decidable then (1) the relation $Q( a) \rightleftharpoons$ "there is an infinite branch in the tree $Tr( a,x,y)$" is definable in $\mathcal M$, and (2) if there is an infinite branch in the tree $Tr( a,x,y)$, then there is a definable in $\mathcal M$ infinite branch.
Svenonius theorem reduces the problem of first-order definability to the problem of relationship between groups of permutations. In the present paper we use this approach to describe the lattice of definable relations for the structure of integer numbers with the successor relation.
In the present paper we discuss the lattice of reducts of $\langle \mathbb{Q}, \{$+$\} \rangle$
The Svenonius theorem describes the (first-order) definability in a structure in terms of permutations preserving the relations of elementary extensions of the structure. In the present paper we prove a version of this theorem using permutations of sequences over the original structure (these are permutations of sequences of tuples of the structure elements as well). We say that such a permutation $φ$ almost preserves a relation if for every sequence of its arguments the value of the relation on an $n$-th element of the sequence and on its image under $φ$ coincide for almost all numbers $n.$ We prove that a relation is definable in a structure iff the relation is almost preserved by all permutations almost preserving the relations of the structure. This version limits consideration to the original structure only and does not refer to any logical notion, such as "elementary equivalence".