arXiv subjects

Evelina Daniyarova

Publications and source records attributed to Evelina Daniyarova.

5 recordsLinked to original sources

arXiv2026

Theory of Interpretations I. Foundations

This is the first paper in a series in which we lay down the foundations of the theory of interpretations. We systematically study different types of interpretations and their properties. Some of these interpretations are known, while others are new. Each of them serves different purposes. In the last section, we describe applications of interpretations to Diophantine problems, first-order classification, isotypeness, definability of structures by types, elimination of imaginaries, richness, logical categories, and bi-interpretations with Z or N. Additionally, throughout the text, we pose some open questions that naturally arise in this context and provide the most typical examples, usually from algebra. The current literature is plagued by discrepancies and inconsistencies in definitions, concepts, and fundamental applications of interpretations. To address this, we thoroughly examine various principal notions, definitions, and arguments, bringing order to the existing theory. Simultaneously, we develop several key concepts, such as regular interpretations, regular bi-interpretations, and invertible interpretations, and outline their main applications.

math.LO↗

arXiv2026

Theory of Interpretations II. Categorical equivalence of projective logical geometries

We introduce projective logical geometry and prove that two algebraic structures are strongly bi-interpretable if and only if their categories of projective logical sets are equivalent relative to the class of interpretation functors, which is also equivalent to their categories of projective definable sets being equivalent relative to the class of translation functors. These constructions generalize two ideas of Boris Plotkin: the concept of geometric equivalence in universal algebraic geometry and the transition from universal algebraic geometry to logical geometry. Furthermore, our categorical approach offers a fresh perspective on the theory of interpretations, enabling us to establish a series of fundamental results using categorical methods.

math.LO↗

arXiv2024

Groups elementarily equivalent to metabelian Baumslag$-$Solitar groups and regular bi-interpretability

We prove that metabelian Baumslag$-$Solitar group $BS(1,k)$, $k>1$, is (strongly) regularly bi-interpretable with the ring of integers $\mathbb{Z}$, and describe in algebraic terms all groups that are elementarily equivalent to $BS(1,k)$.

math.GR↗

arXiv2011

Algebraic geometry over algebraic structures II: Foundations

In this paper we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove Unification Theorems which gather the description of coordinate algebras by several ways.

math.AG↗

arXiv2010

Algebraic geometry over algebraic structures III: Equationally Noetherian property and compactness

In this paper we discuss some special generalizations of equationally Noetherian property which naturally arise in the universal algebraic geometry. We introduce weakly equationally Noetherian, qw-compact, uw-compact, and weakly uw-compact algebras and then examine properties of such algebras. Also we consider the connections between five classes: the class of equationally Noetherian algebras, the class of weakly equationally Noetherian algebras, the class of uw-compact algebras, the class of weakly uw-compact algebras, and the class of qw-compact algebras.

math.AG↗