Citizendia
Your Ad Here

In algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Broadly speaking pure mathematics is Mathematics motivated entirely for reasons other than application In Mathematics, a set is said to be closed under some operation if the operation on members of the set produces a member of the set In its simplest meaning in Mathematics and Logic, an operation is an action or procedure which produces a new value from one or more input values In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Abstract algebra is primarily the study of algebraic structures and their properties. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules The notion of algebraic structure has been formalized in universal algebra. In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models"

Abstractly, an "algebraic structure" is the collection of all possible models of a given set of axioms. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models More concretely, an algebraic structure is any particular model of some set of axioms. For example, the monster group both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other groups. In the Mathematical field of Group theory, the Monster group M or F 1 (also known as the Fischer-Griess Monster or the Friendly Giant In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element This article employs both meanings of "structure. "

This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all groups are also semigroups and magmas. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure.

Contents

Structures whose axioms are all identities

Axioms of monoids, groups, abelian groups, rngs, rings, commutative rings and fields.
Axioms of monoids, groups, abelian groups, rngs, rings, commutative rings and fields. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

If the axioms defining a structure are all identities, the structure is a variety (not to be confused with algebraic variety in the sense of algebraic geometry). In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. In Predicate logic, universal quantification is an attempt to formalize the notion that something (a Logical predicate) is true for everything, or every In Mathematical logic, the universe of a structure (or model) is its domain. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. Quantification has two distinct meanings In Mathematics and Empirical science, it refers to human acts known as Counting and Measuring This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations The study of varieties is an important part of universal algebra. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models"

All structures in this section are varieties. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities Some of these structures are most naturally axiomatized using one or more nonidentities, but are nevertheless varieties because there exists an equivalent axiomatization, one perhaps less perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and differ from varieties in their metamathematical properties. In general metamathematics or meta-mathematics is a scientific reflection and Knowledge about mathematics seen as an entity/ object in Human

In this section and the following one, structures are listed in approximate order of increasing complexity, operationalized as follows:

The indentation structure employed in this section and the one following is intended to convey information. If structure B is under structure A and more indented, then all theorems of A are theorems of B; the converse does not hold. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements

Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are linked by the absorption law. In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Algebra, the absorption law is an identity linking a pair of Binary operations Any two Binary operations, say $ and % are subject to Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Simple structures: No binary operation:

Group-like structures:

One binary operation, denoted by concatenation. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two For concatenation of general lists see Append. In Computer programming, string concatenation is the operation of joining two character For monoids, boundary algebras, and sloops, S is a pointed set. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation Laws of Form (hereinafter LoF) is a book by G Spencer-Brown, published in 1969 that straddles the boundary between Mathematics and of In Mathematics, a pointed set is a set X with a distinguished basepoint x 0 in X.

Three binary operations. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. Quasigroups feature 3 binary operations only because establishing the quasigroup cancellation property by means of identities alone requires two binary operations in addition to the group operation. In Mathematics, the notion of cancellative is a generalization of the notion of Invertible.

Lattice: Two or more binary operations, including meet and join, connected by the absorption law. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In mathematics a meet on a set is defined either as the unique Infimum (greatest lower bound with respect to a Partial order on the set provided an infimum exists In mathematical Order theory, join is a Binary operation on a Partially ordered set that gives the Supremum (least upper bound of its arguments In Algebra, the absorption law is an identity linking a pair of Binary operations Any two Binary operations, say $ and % are subject to S is both a meet and join semilattice, and is a pointed set if and only if S is bounded. In mathematics a meet on a set is defined either as the unique Infimum (greatest lower bound with respect to a Partial order on the set provided an infimum exists In mathematical Order theory, join is a Binary operation on a Partially ordered set that gives the Supremum (least upper bound of its arguments In Mathematics, a pointed set is a set X with a distinguished basepoint x 0 in X. Lattices often have no unary operations. Every true statement has a dual, obtained by replacing every instance of meet with join, and vice versa.

Ringoids: Two binary operations, addition and multiplication, with multiplication distributing over addition. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Addition is the mathematical process of putting things together In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law Semirings are pointed sets. In Mathematics, a pointed set is a set X with a distinguished basepoint x 0 in X.

N. B. The above definition of ring does not command universal assent. Some authorities employ "ring" to denote what is here called a rng, and refer to a ring in the above sense as a "ring with identity. "

Modules: Composite Systems Defined over Two Sets, M and R: The members of:

  1. R are scalars, denoted by Greek letters. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication R is a ring under the binary operations of scalar addition and multiplication;
  2. M are module elements (often but not necessarily vectors), denoted by Latin letters. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added M is an abelian group under addition. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the There may be other binary operations. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

The scalar product of scalars and module elements is a function RxM→M which commutes, associates (∀r,s∈R, ∀x∈M, r(sx) = (rs)x ), has 1 as identity element, and distributes over module and scalar addition. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R If only the pre(post)multiplication of module elements by scalars is defined, the result is a left (right) module.

Vector spaces, closely related to modules, are defined in the next section. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

Structures with some axioms that are not identities

The structures in this section are not varieties because they cannot be axiomatized with identities alone. In Universal algebra, a variety of algebras is the class of all Algebraic structures of a given signature satisfying a given set of identities Nearly all of the nonidentities below are one of two very elementary kinds:

  1. The starting point for all structures in this section is a "nontrivial" ring, namely one such that S≠{0}, 0 being the additive identity element. In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that The nearest thing to an identity implying S≠{0} is the nonidentity 0≠1, which requires that the additive and multiplicative identities be distinct.
  2. Nearly all structures described in this section include identities that hold for all members of S except 0. In order for an algebraic structure to be a variety, its operations must be defined for all members of S; there can be no partial operations.

Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e. g. , fields and vector spaces. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added Moreover, much of theoretical physics can be recast as models of multilinear algebras. In Mathematics, multilinear algebra extends the methods of Linear algebra. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not have. For example, neither the product of integral domains nor a free field over any set exist. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such

Arithmetics: Two binary operations, addition and multiplication. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two S is an infinite set. In Set theory, an infinite set is a set that is not a Finite set. Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0. In Mathematics, a unary operation is an operation with only one Operand, i The primitive recursive functions are defined using primitive recursion and composition as central operations and are a strict Subset of the recursive

Field-like structures: Two binary operations, addition and multiplication. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two S is nontrivial, i. e. , S≠{0}.

The following structures are not varieties for reasons in addition to S≠{0}:

Composite Systems: Vector Spaces, and Algebras over Fields. Two Sets, M and R, and at least three binary operations.

The members of:

  1. M are vectors, denoted by lower case letters. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added M is at minimum an abelian group under vector addition, with distinguished member 0. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the
  2. R are scalars, denoted by Greek letters. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication R is a field, nearly always the real or complex field, with 0 and 1 as distinguished members. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

Three binary operations.

Four binary operations.

Composite Systems: Multilinear algebras. In Mathematics, multilinear algebra extends the methods of Linear algebra. Two sets, V and K. Four binary operations:

  1. The members of V are multivectors (including vectors), denoted by lower case Latin letters. In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two In a Grassmann algebra, a multivector is an element of a vector space V. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added V is an abelian group under multivector addition, and a monoid under outer product. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In a Grassmann algebra, a multivector is an element of a vector space V. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Linear algebra, the outer product typically refers to the tensor product of two vectors. The outer product goes under various names, and is multilinear in principle but usually bilinear. In Mathematics, the modern Component-free approach to the theory of Tensors views tensors initially as Abstract objects expressing some definite type of The outer product defines the multivectors recursively starting from the vectors. Thus the members of V have a "degree" (see graded algebra below). In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure Multivectors may have an inner product as well, denoted u•v: V×V→K, that is symmetric, linear, and positive definite; see inner product space above. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, the term positive-definite function may refer to a couple of different concepts In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.
  2. The properties and notation of K are the same as those of R above, except that K may have -1 as a distinguished member. K is usually the real field, as multilinear algebras are designed to describe physical phenomena without complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted
  3. The multiplication of scalars and multivectors, V×K→V, has the same properties as the multiplication of scalars and module elements that is part of a module. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars

Examples

Some recurring universes: N=natural numbers; Z=integers; Q=rational numbers; R=real numbers; C=complex numbers. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

N is a pointed unary system, and under addition and multiplication, is both the standard interpretation of Peano arithmetic and a commutative semiring. In Mathematical logic, the Peano axioms, also known as the Dedekind-Peano axioms or the Peano postulates, are a set of Axioms for the Natural In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse

Boolean algebras are at once semigroups, lattices, and rings. In Abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real They would even be Abelian groups if the identity and inverse elements were identical instead of complements. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the

Group-like structures

Lattices

Ring-like structures

Integral domains

Fields

Allowing additional structure

Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The added structure must be compatible, in some sense, with the algebraic structure.

Category theory

The discussion above has been cast in terms of elementary abstract and universal algebra. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" Category theory is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets A category is a collection of objects with associated morphisms. Every algebraic structure has its own notion of homomorphism, namely any function compatible with the operation(s) defining the structure. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In this way, every algebraic structure gives rise to a category. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function This concrete category may be seen as a category of sets with added category-theoretic structure. In Mathematics, a concrete category is commonly understood as a category whose objects are structured sets, whose Morphisms are structure-preserving In Mathematics, the category of sets, denoted as Set, is the category whose objects are all sets and whose Morphisms are In Mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the category of topological spaces, often denoted Top, is the category whose objects are Topological spaces and whose

There are various concepts in category theory that try to capture the algebraic character of a context, for instance

See also

References

A monograph available free online:

Category theory:

External links

Dictionary

algebraic structure

-noun

  1. (algebra) One or more sets together with a set of operations such that the sets are closed under the operations and are satisfying some axioms.
© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic