Related papers: Varieties
We formalize the general principle of significance with respect to binary relations which is a universal tool for description and analysis of various situations in and apart from mathematics. We derive the basic properties and focus on a…
We introduce a class of real algebraic varieties characterised by a simple rationality condition, which exhibit strong properties regarding approximation of continuous and smooth mappings by regular ones. They form a natural counterpart to…
This article is an introduction to formal languages from the point of view of combinatorial group theory. Group theoretic applications are included and language classes are defined algebraically.
Classes of algebraic structures that are defined by equational laws are called varieties or equational classes. A variety is finitely generated if it is defined by the laws that hold in some fixed finite algebra. We show that every…
The aim of these notes is to provide a succinct, accessible introduction to some of the basic ideas of category theory and categorical logic. The notes are based on a lecture course given at Oxford over the past few years. They contain…
We give an introduction to the topics of our forthcoming work, in which we introduce and study new mathematical objects which we call "higher theories" of algebras, where inspiration for the term comes from William Lawvere's notion of…
In this paper, we introduce the notion of relation type of analytic and formal algebras and prove that it is well-defined and invariant by describing this notion in terms of the Andr\'e-Quillen homology and using the Jacobi-Zariski long…
We give a short appreciation of Mumford's work on the moduli of varieties by putting it into historical context. By reviewing earlier works we highlight the innovations introduced by Mumford. Then we discuss recent developments whose…
We define a general class of dependent type theories, encompassing Martin-L\"of's intuitionistic type theories and variants and extensions. The primary aim is pragmatic: to unify and organise their study, allowing results and constructions…
In these self-contained low prerequisite introductory notes we first present (in part 1) basic concepts of set theory and algebra without explicit category theory. We then present (in part 2) basic category theory involving a somewhat…
In the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations,…
The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine…
We show that many important varieties and sets of varieties of semigroups may be defined by relatively simple and transparent first-order formulas in the lattice of all semigroup varieties.
We introduce a class of algebras that can be used as recognisers for regular tree languages. We show that it is the only such class that forms a pseudo-variety and we prove the existence of syntactic algebras. Finally, we give a more…
This paper proposes a way to compute the meanings associated with sentences with generic noun phrases corresponding to the generalized quantifier most. We call these generics specimens and they resemble stereotypes or prototypes in lexical…
The Eilenberg correspondence relates varieties of regular languages to pseudovarieties of finite monoids. Various modifications of this correspondence have been found with more general classes of regular languages on one hand and classes of…
We study fundamental forms of algebraic varieties using the sheaves of principal parts of line bundles and establish a vanishing theorem for any order fundamental forms. We also give connection of fundamental forms with the higher order…
We propose the use of algebras of generalized functions for the analysis of certain highly singular problems in the calculus of variations. After a general study of extremal problems on open subsets of Euclidean space in this setting we…
A general simplicity problem in category theory is proposed. A particular example, the simplest choice of generators of an algebra is specified and illustrated by an example.
We begin with a short exposition of the theory of lattice varieties. This includes a description of their orbit structure and smooth locus. We construct a flat cover of the lattice variety and show that it is a complete intersection. We…