Algebraic structure

In, an algebraic structure on a A (called the underlying set, carrier set or domain) is a collection of s on A of finite , together with a finite set of , called s of the structure that these operations must satisfy. In the context of, the set A with this is called an algebra, while, in other contexts, it is (somewhat ambiguously) called an algebraic structure, the term algebra being reserved for specific algebraic structures that are s over a  or  over a.

Examples of algebraic structures include, , , and. More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include s,, and.

The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in. The language of is used to express and study relationships between different classes of algebraic and non-algebraic objects. This is because it is sometimes possible to find strong connections between some classes of objects, sometimes of different kinds. For example, establishes a connection between certain fields and groups: two algebraic structures of different kinds.

Introduction
Addition and multiplication on numbers are the prototypical example of an operation that combines two elements of a set to produce a third. These operations obey several algebraic laws. For example, a + (b + c) = (a + b) + c and a(bc) = (ab)c, both examples of the associative law. Also a + b = b + a, and ab = ba, the commutative law. Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, rotations of objects in three-dimensional space can be combined by performing the first rotation and then applying the second rotation to the object in its new orientation. This operation on rotations obeys the associative law, but can fail the commutative law.

Mathematicians give names to sets with one or more operations that obey a particular collection of laws, and study them in the abstract as algebraic structures. When a new problem can be shown to follow the laws of one of these algebraic structures, all the work that has been done on that category in the past can be applied to the new problem.

In full generality, algebraic structures may involve an arbitrary number of sets and operations that can combine more than two elements (higher ), but this article focuses on binary operations on one or two sets. The examples here are by no means a complete list, but they are meant to be a representative list and include the most common structures. Longer lists of algebraic structures may be found in the external links and within . Structures are listed in approximate order of increasing complexity.

One set with operations
Simple structures: no :


 * : a degenerate algebraic structure S  having no operations.
 * : S has one or more distinguished elements, often 0, 1, or both.
 * Unary system: S and a single over S.
 * Pointed unary system: a unary system with S a pointed set.

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.


 * : S and a single binary operation over S.
 * : an magma.
 * : a semigroup with.
 * : a monoid with a unary operation (inverse), giving rise to s.
 * : a group whose binary operation is.
 * : a semigroup whose operation is and commutative. The binary operation can be called either  or.
 * : a magma obeying the . A quasigroup may also be represented using three binary operations.


 * : a quasigroup with.

Ring-like structures or Ringoids: two binary operations, often called and, with multiplication  over addition.


 * : a ringoid such that S is a monoid under each operation. Addition is typically assumed to be commutative and associative, and the monoid product is assumed to distribute over the addition on both sides, and the additive identity 0 is an in the sense that 0&#8239;x = 0 for all x.
 * : a semiring whose additive monoid is a (not necessarily abelian) group.
 * : a semiring whose additive monoid is an abelian group.
 * : a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the rather than associativity.
 * : a commutative ring with idempotent multiplication operation.
 * : a commutative ring which contains a multiplicative inverse for every nonzero element.
 * s: a semiring with idempotent addition and a unary operation, the, satisfying additional properties.
 * : a ring with an additional unary operation (*) satisfying additional properties.

Lattice structures: two or more binary operations, including operations called, connected by the.


 * : a lattice in which arbitrary s exist.
 * : a lattice with a and least element.
 * : a bounded lattice with a unary operation, complementation, denoted by undefined. The join of an element with its complement is the greatest element, and the meet of the two elements is the least element.
 * : a lattice whose elements satisfy the additional modular identity.
 * : a lattice in which each of meet and join over the other. Distributive lattices are modular, but the converse does not hold.
 * : a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation. This can be shown to be equivalent with the ring-like structure of the same name above.
 * : a bounded distributive lattice with an added binary operation,, denoted by →, and governed by the axioms x&#8239;→&#8239;x = 1, x&#8239;(x&#8239;→&#8239;y) = x&#8239;y, y&#8239;(x&#8239;→&#8239;y) = y, x&#8239;→&#8239;(y&#8239;z) = (x&#8239;→&#8239;y)&#8239;(x&#8239;→&#8239;z).

Arithmetics: two s, addition and multiplication. S is an. Arithmetics are pointed unary systems, whose is , and with distinguished element 0.


 * . Addition and multiplication are defined by means of successor. 0 is the identity element for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to Peano arithmetic.
 * . Robinson arithmetic with an of . Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.

Two sets with operations
-like structures: composite systems involving two sets and employing at least two binary operations.


 * : a group G with a set Ω and a binary operation Ω&#8239;×&#8239;G → G satisfying certain axioms.
 * : an abelian group M and a ring R acting as operators on M. The members of R are sometimes called s, and the binary operation of scalar multiplication is a function R&#8239;×&#8239;M → M, which satisfies several axioms. Counting the ring operations these systems have at least three operations.
 * : a module where the ring R is a or.
 * : a vector space with a decomposition breaking the space into "grades".
 * : a vector space V over a field F with a function from V into F satisfying certain properties. Every quadratic space is also an inner product space (see below).

-like structures: composite system defined over two sets, a ring R and an R-module M equipped with an operation called multiplication. This can be viewed as a system with five binary operations: two operations on R, two on M and one involving both R and M.


 * (also R-algebra): a module over a R, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and  with respect to multiplication by elements of R. The theory of an  is especially well developed.
 * : an algebra over a ring such that the multiplication is.
 * : a module over a commutative ring, equipped with a ring multiplication operation that is not necessarily associative. Often associativity is replaced with a different identity, such as, the , or the.
 * : a vector space with a "comultiplication" defined dually to that of associative algebras.
 * : a special type of nonassociative algebra whose product satisfies the.
 * : a vector space with a "comultiplication" defined dually to that of Lie algebras.
 * : a graded vector space with an algebra structure compatible with the grading. The idea is that if the grades of two elements a and b are known, then the grade of ab is known, and so the location of the product ab is determined in the decomposition.
 * : an F vector space V with a sesquilinear binary operation V × V → F.

Four or more binary operations:


 * : an associative algebra with a compatible coalgebra structure.
 * : a Lie algebra with a compatible bialgebra structure.
 * : a bialgebra with a connection axiom (antipode).
 * : a graded associative algebra equipped with an from which may be derived several possible inner products.  s and s are special cases of this construction.

Hybrid structures
Algebraic structures can also coexist with added structure of non-algebraic nature, such as or a. The added structure must be compatible, in some sense, with the algebraic structure.


 * : a group with a topology compatible with the group operation.
 * : a topological group with a compatible smooth structure.
 * s, s and s: each type of structure with a compatible.
 * : a linearly ordered group for which the holds.
 * : a vector space whose M has a compatible topology.
 * : a vector space with a compatible . If such a space is (as a metric space) then it is called a.
 * : an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.
 * : a *-algebra of operators on a Hilbert space equipped with the.
 * : a *-algebra of operators on a Hilbert space equipped with the.

Universal algebra
Algebraic structures are defined through different configurations of s. abstractly studies such objects. One major dichotomy is between structures that are axiomatized entirely by identities and structures that are not. If all axioms defining a class of algebras are identities, then this class is a (not to be confused with  of ).

Identities are equations formulated using only the operations the structure allows, and variables that are tacitly over the relevant. Identities contain no, , or of any kind other than the allowed operations. The study of varieties is an important part of. An algebraic structure in a variety may be understood as the of term algebra (also called "absolutely  ") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given generate a free algebra, the  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 an operator that takes zero arguments. Given a (countable) set of variables x, y, z, etc. the term algebra is the collection of all possible 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. The axioms can be represented as trees. These equations induce es on the free algebra; the quotient algebra then has the algebraic structure of a group.

Some structures do not form varieties, because either:


 * 1) It is necessary that 0 ≠ 1, 0 being the additive  and 1 being a multiplicative identity element, but this is a nonidentity;
 * 2) Structures such as fields have some axioms that hold only for nonzero members of S. 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., s and s. Structures with nonidentities present challenges varieties do not. For example, the of two s is not a field, because $$(1,0)\cdot(0,1)=(0,0)$$, but fields do not have s.

Category theory
is another tool for studying 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, namely any compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a. For example, the has all  as objects and all s as morphisms. This may be seen as a  with added category-theoretic. Likewise, the category of s (whose morphisms are the continuous group homomorphisms) is a with extra structure. A 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


 * functors and categories
 * functors and categories
 * functors and categories
 * functors and categories
 * functors and categories

Different meanings of "structure"
In a slight, the word "structure" can also refer to just the operations on a structure, instead of the underlying set itself. For example, the sentence, "We have defined a ring structure on the set $$A$$," means that we have defined operations on the set $$A$$. For another example, the group $$(\mathbb Z, +)$$ can be seen as a set $$\mathbb Z$$ that is equipped with an algebraic structure, namely the operation $$+$$.