arXiv subjects

Evgueni Vassiliev

Publications and source records attributed to Evgueni Vassiliev.

3 recordsLinked to original sources

arXiv2026

Dense-codense expansions of quasiminimal pregeometry structures

We study expansions of quasiminimal pregeometry structures with a dense codense unary predicate and their relation with the complexity properties of the pregeometry of the underlying structure. We consider beautiful pairs as well as $H$-structures. We show each of these expansions can be axiomatized with a single $L_{ω_1 ω}(Q)$-sentence and that both expansions are $ω$-stable. For $H$-structures we provide a natural notion of independence in the expansion and when the underlying structure is modular, we also provide a natural notion of independence for beautiful pairs. Then we relate the complexity of the pregeometry to properties of the expansions.

math.LO↗

arXiv2023

Vector spaces with a dense-codense generic submodule

We study expansions of a vector space $V$ over a field $\mathbb F$, possibly with extra structure, with a generic submodule over a subring of $\mathbb F$. We construct a natural expansion by existentially defined functions so that the expansion in the extended language satisfies quantifier elimination. We show that this expansion preserves tame model theoretic properties such as stability, NIP, NTP$_1$, NTP$_2$ and NSOP$_1$. We also study induced independence relations in the expansion.

math.LO↗

arXiv2018

Supersimple structures with a dense independent subset

Based on the work done in \cite{BV-Tind,DMS} in the o-minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking-independent elements that is dense inside a partial type $\mathcal{G}(x)$, which we call $H$-structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again $H$-structures. We prove that under these assumptions the expansion is supersimple and characterize forking and canonical bases of types in the expansion. We also analyze the effect these expansions have on one-basedness and CM-triviality. In the one-based case, when $T$ has $SU$-rank $ω^α$ and the $SU$-rank is continuous, we take $\mathcal{G}(x)$ to be the type of elements of $SU$-rank $ω^α$ and we describe a natural "geometry of generics modulo $H$" associated with such expansions and show it is modular.

math.LO↗