In mathematics, and more specifically abstract algebra, the term algebraic structure generally refers to an arbitrary set with one or more binary operations defined on it. This idea effectively brings out the algebraic properties of the members of the set, and in turn defines an overall structure. Common examples of structures include groups, rings, fields and lattices. More complex algebraic structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex structures include vector spaces, modules and algebras.
The main motivation behind the study of algebraic structures is that many seemingly unrelated concepts can be related in terms of their algebraic properties. This provides links between them and gives a more in depth understanding of the mathematics involved. Although algebraic structures are mainly studied in pure mathematics, they do have applications in other fields, for example mathematical physics.
The properties of specific algebraic structures are studied in the branch known as abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. Category theory is used to study the relationships between two or more classes of algebraic structures, often of different kinds. For example, Galois theory studies the connection between certain fields and groups, algebraic structures of two different kinds.
In a slight abuse of notation, the expression "structure" can also refer only to the operations on a structure, and not to the underlying set itself. For example, the group can be seen as a set which is equipped with an "algebraic structure", namely the operation .
Algebraic structures |
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Universal algebra often considers classes of algebraic structures (such as the class of all groups), together with operations (such as products) and relations (such as "substructure") between these algebras. These classes are usually defined by "axioms", that is, a list of properties that all these structures have to share. If all axioms defining a class of algebras are "identities" , then the corresponding class is called variety (not to be confused with algebraic variety in the sense of algebraic geometry).
Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra.
An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure E. The quotient algebra T/E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered to be an operator taking zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all possible terms involving m, i, e and the variables; so for example, m(i(x), m(x,m(y,e))) would be an element of the term algebra. One of the axioms defining a group is the identity m(x, i(x)) = e; another is m(x,e) = x. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.
All structures in this section are elements of naturally defined varieties. 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 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.
Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. 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. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models.
Simple structures: No binary operation:
Group-like structures:
One binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. For nonassociative operations, it becomes necessary to indicate the order of operations with parentheses. For monoids, boundary algebras, and sloops, S is a pointed set.
Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. 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.
Lattice: Two or more binary operations, including meet and join, connected by the absorption law. S is both a meet and join semilattice, and is a pointed set if and only if S is bounded. 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. Semirings are pointed sets.
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:
The scalar multiplication 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. 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.
The structures in this section are not axiomatized with identities alone, so the classes considered below are not varieties. Nearly all of the nonidentities below are one of two very elementary kinds:
Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and vector spaces. Moreover, much of theoretical physics can be recast as models of multilinear algebras. 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.
Arithmetics: Two binary operations, addition and multiplication. S is an infinite set. Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0.
Field-like structures: Two binary operations, addition and multiplication. 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:
Three binary operations.
Four binary operations.
Composite Systems: Multilinear algebras. Two sets, V and K. Four binary operations:
Some recurring universes: N= natural numbers; Z= integers; Q= rational numbers; R= real numbers; C= complex numbers.
N is a pointed unary system, and under addition and multiplication, is both the standard interpretation of Peano arithmetic and a commutative semiring.
Boolean algebras are at once semigroups, lattices, and rings. They would even be abelian groups if the identity and inverse elements were identical instead of complements.
Group-like structures
Ring-like structures
Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a topology. The added structure must be compatible, in some sense, with the algebraic structure.
The discussion above has been cast in terms of elementary abstract and universal algebra. Category theory is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). 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 this way, every algebraic structure gives rise to a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure.
There are various concepts in category theory that try to capture the algebraic character of a context, for instance
A monograph available online:
Category theory:
In mathematics, and more specifically abstract algebra, the term algebraic structure generally refers to an arbitrary set with one or more binary operations defined on it. This idea effectively brings out the algebraic properties of the members of the set, and in turn defines an overall structure. Common examples of structures include groups, rings, fields and lattices. More complex algebraic structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex structures include vector spaces, modules and algebras.
The main motivation behind the study of algebraic structures is that many seemingly unrelated concepts can be related in terms of their algebraic properties. This provides links between them and gives a more in depth understanding of the mathematics involved. Although algebraic structures are mainly studied in pure mathematics, they do have applications in other fields, for example mathematical physics.
The properties of specific algebraic structures are studied in the branch known as abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. Category theory is used to study the relationships between two or more classes of algebraic structures, often of different kinds. For example, Galois theory studies the connection between certain fields and groups, algebraic structures of two different kinds.
In a slight abuse of notation, the expression "structure" can also refer only to the operations on a structure, and not to the underlying set itself. For example, the group can be seen as a set which is equipped with an "algebraic structure", namely the operation .
Algebraic structures |
---|
Universal algebra often considers classes of algebraic structures (such as the class of all groups), together with operations (such as products) and relations (such as "substructure") between these algebras. These classes are usually defined by "axioms", that is, a list of properties that all these structures have to share. If all axioms defining a class of algebras are "identities" , then the corresponding class is called variety (not to be confused with algebraic variety in the sense of algebraic geometry).
Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant universe. Identities contain no connectives, existentially quantified variables, or relations of any kind other than the allowed operations. The study of varieties is an important part of universal algebra.
An algebraic structure in a variety may be understood as the quotient algebra of term algebra (also called "absolutely free algebra") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given signatures generate a free algebra, the term algebra T. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure E. The quotient algebra T/E is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator m, taking two arguments, and the inverse operator i, taking one argument, and the identity element e, a constant, which may be considered to be an operator taking zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all possible terms involving m, i, e and the variables; so for example, m(i(x), m(x,m(y,e))) would be an element of the term algebra. One of the axioms defining a group is the identity m(x, i(x)) = e; another is m(x,e) = x. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.
All structures in this section are elements of naturally defined varieties. 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 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.
Ringoids and lattices can be clearly distinguished despite both having two defining binary operations. 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. Ringoids also tend to have numerical models, while lattices tend to have set-theoretic models.
Simple structures: No binary operation:
Group-like structures:
One binary operation. The binary operation can be indicated by any symbol, or with no symbol (juxtaposition) as is done for ordinary multiplication of real numbers. For nonassociative operations, it becomes necessary to indicate the order of operations with parentheses. For monoids, boundary algebras, and sloops, S is a pointed set.
Three binary operations. Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative) magmas. 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.
Lattice: Two or more binary operations, including meet and join, connected by the absorption law. S is both a meet and join semilattice, and is a pointed set if and only if S is bounded. 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. Semirings are pointed sets.
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:
The scalar multiplication 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. 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.
The structures in this section are not axiomatized with identities alone, so the classes considered below are not varieties. Nearly all of the nonidentities below are one of two very elementary kinds:
Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and vector spaces. Moreover, much of theoretical physics can be recast as models of multilinear algebras. 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.
Arithmetics: Two binary operations, addition and multiplication. S is an infinite set. Arithmetics are pointed unary systems, whose unary operation is injective successor, and with distinguished element 0.
Field-like structures: Two binary operations, addition and multiplication. 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:
Three binary operations.
Four binary operations.
Composite Systems: Multilinear algebras. Two sets, V and K. Four binary operations:
Some recurring universes: N= natural numbers; Z= integers; Q= rational numbers; R= real numbers; C= complex numbers.
N is a pointed unary system, and under addition and multiplication, is both the standard interpretation of Peano arithmetic and a commutative semiring.
Boolean algebras are at once semigroups, lattices, and rings. They would even be abelian groups if the identity and inverse elements were identical instead of complements.
Group-like structures
Ring-like structures
Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a topology. The added structure must be compatible, in some sense, with the algebraic structure.
The discussion above has been cast in terms of elementary abstract and universal algebra. Category theory is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). 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 this way, every algebraic structure gives rise to a category. For example, the category of groups has all groups as objects and all group homomorphisms as morphisms. This concrete category may be seen as a category of sets with added category-theoretic structure. Likewise, the category of topological groups (whose morphisms are the continuous group homomorphisms) is a category of topological spaces with extra structure. A forgetful functor between categories of algebraic structures "forgets" a part of a structure.
There are various concepts in category theory that try to capture the algebraic character of a context, for instance
A monograph available online:
Category theory: