Variety (universal algebra)

In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of , a variety of algebras, together with its homomorphisms, forms a ; these are usually called finitary algebraic categories. A covariety is the class of all coalgebraic structures of a given signature. A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial equations. They are formally quite distinct and their theories have little in common. The term "variety of algebras" refers to algebras in the general sense of universal algebra; there is also a more specific sense of algebra, namely as algebra over a field, i.e. a vector space equipped with a bilinear multiplication. A signature (in this context) is a set, whose elements are called operations, each of which is assigned a natural number (0, 1, 2,...) called its arity. Given a signature and a set , whose elements are called variables, a word is a finite planar rooted tree in which each node is labelled by either a variable or an operation, such that every node labelled by a variable has no branches away from the root and every node labelled by an operation has as many branches away from the root as the arity of . An equational law is a pair of such words; the axiom consisting of the words and is written as . A theory consists of a signature, a set of variables, and a set of equational laws. Any theory gives a variety of algebras as follows. Given a theory , an algebra of consists of a set together with, for each operation of with arity , a function such that for each axiom and each assignment of elements of to the variables in that axiom, the equation holds that is given by applying the operations to the elements of as indicated by the trees defining and .

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (32)

Related people (1)

Related units (1)

Related concepts (16)

Related courses (4)

Related lectures (32)

We extend the group-theoretic notion of conditional flatness for a localization functor to any pointed category, and investigate it in the context of homological categories and of semi-abelian categories. In the presence of functorial fiberwise localizatio ...

CAMBRIDGE UNIV PRESS2023

In this thesis, we study interactions between algebraic and coalgebraic structures in infinity-categories (more precisely, in the quasicategorical model of (infinity, 1)-categories). We define a notion of a Hopf algebra H in an E-2-monoidal infinity-catego ...

EPFL2022

Development of remote C-H bond functionalization of redox-active alcohol derivatives and oxidative transformations under photoredox catalysis

Alexandre Julien Florent Leclair

The development of radical-based remote C(sp3)-H functionalizations of redox-active alcohol derivatives, oxidative cycloaddition and rearrangement transformations under visible light photocatalysis constitute the basis of this thesis.The first chapter desc ...

EPFL2021

Abstract algebra

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.

Monad (category theory)

In , a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a in the of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.

Free object

In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set A can be thought of as being a "generic" algebraic structure over A: the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices. The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations).

Group representation theory studies the actions of groups on vector spaces. This allows the use of linear algebra to study certain group theoretical questions. In this course the groups in question wi

Après une introduction à la théorie des catégories, nous appliquerons la théorie générale au cas particulier des groupes, ce qui nous permettra de bien mettre en perspective des notions telles que quo

The aim of the course is to give an introduction to linear algebraic groups and to give an insight into a beautiful subject that combines algebraic geometry with group theory.

Morphisms between Affine Varieties MATH-328: Algebraic geometry I - Curves

Covers defining morphisms between affine algebraic varieties and constructing morphisms using algebraic homomorphisms.

Monads: Data Structures with Laws CS-210: Functional programming

Covers monads as data structures with specific laws and examples like List and Option.

Algebraic Structures of Morphism Spaces MATH-110(a): Advanced linear algebra I

Explores algebraic structures of morphism spaces, including stability properties and injective ring homomorphisms.