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Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a…
Equational Artinian algebras were introduced in our previous work: {\em Equational conditions in universal algebraic geometry, to appear in Algebra and Logic, 2015}. In this note, we define the notion of {\em radical topology with respect…
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…
We study equations over boolean algebras with distinguished elements. We prove the criteria, when a boolean algebra is equationally Noetherian, weakly equationally Noetherian, $\mathbf{q}_\omega$-compact or $\mathbf{u}_\omega$-compact. Also…
In this note, we give a new characterization for an algebra to be $\qo$-compact in terms of {\em super-product operations} on the lattice of congruences of the relative free algebra.
We use tools of mathematical logic to analyse the notion of a path on an complex algebraic variety, and are led to formulate a "rigidity" property of fundamental groups specific to algebraic varieties, as well as to define a bona fide…
We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology…
A number of geometric properties of $\Omega$-groups from a given variety of $\Omega$-groups can be characterized using the notions of domain and equational domain. An $\Omega$-group $H$ of a variety $\Theta$ is an equational domain in…
This short survey article reviews current understand- ing of the structure of noetherian Hopf algebras. The focus is on homological properties. A number of open problems are listed.
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.
A brief survey of some aspects of noetherian Hopf algebras is given, concentrating on structure, homology, and classification, and accompanied by a panoply of open problems.
We review the relation between homotopy algebras of conformal field theory and geometric structures arising in sigma models. In particular we formulate conformal invariance conditions, which in the quasi-classical limit are Einstein…
We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.
This note surveys basic topological properties of nonarchimedean analytic spaces, in the sense of Berkovich, including the recent tameness results of Hrushovski and Loeser. We also discuss interactions between the topology of nonarchimedean…
The algebras for all possible Lorentzian and Euclidean kinematics with $\frak{so}(3)$ isotropy except static ones are re-classified. The geometries for algebras are presented by contraction approach. The relations among the geometries are…
This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications.
A brief introduction to universal algebra and the theory of topological algebras, their varieties, and free topological algebras is presented. Free topological Mal'tsev algebras are studied. Their properties, relationship with topological…
Uniform interpolation properties are defined for equational consequence in a variety of algebras and related to properties of compact congruences on first the free and then the finitely presented algebras of the variety. It is also shown,…
We describe the mathematical properties of pairwise comparisons matrices with coefficients in an arbitrary group. We provide a vocabulary adapted for the description of main algebraic properties of inconsistency maps, describe an example…
In this article, a new and natural topology on the prime spectrum is established which behaves completely as the dual of the Zariski topology. It is called the flat topology. The basic and also some sophisticated properties of the flat…