Algebraic structure

In mathematics, an algebraic structure consists of a nonempty set "A" (called the underlying set, carrier set or domain), a collection of operations on "A" of finite arity (typically binary operations), and a finite set of identities, known as axioms, that these operations must satisfy.
An algebraic structure may be based on other algebraic structures with operations and axioms involving several structures. For instance, a vector space involves a second structure called a field, and an operation called "scalar multiplication" between elements of the field (called "scalars"), and elements of the vector space (called "vectors").
In the context of universal algebra, the set "A" with this structure 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 vector spaces over a field or modules over a commutative ring.
The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. The language of category theory 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, Galois theory establishes a connection between certain fields and groups: two algebraic structures of different kinds.
Introduction.
Addition and multiplication of real numbers are the prototypical examples of operations that combine two elements of a set to produce a third element of the set. These operations obey several algebraic laws. For example, "a" + ("b" + "c") = ("a" + "b") + "c" and "a"("bc") = ("ab")"c" as the "associative laws". Also "a" + "b" = "b" + "a" and "ab" = "ba" as the "commutative laws." Many systems studied by mathematicians have operations that obey some, but not necessarily all, of the laws of ordinary arithmetic. For example, rotations of an object in three-dimensional space can be combined by, for example, performing the first rotation on the object and then applying the second rotation on it in its new orientation made by the previous rotation. Rotation as an operation obeys the associative law, but can fail to satisfy 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 collection of operations, including operations that combine more than two elements (higher arity operations) and operations that take only one argument (unary operations). The examples used 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.
Examples.
One set with operations.
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.
Ring-like structures or Ringoids: two binary operations, often called addition and multiplication, with multiplication distributing over addition.
Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law.
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.
Two sets with operations.
Module-like structures: composite systems involving two sets and employing at least two binary operations.
Algebra-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".
Four or more binary operations:
Hybrid structures.
Algebraic structures can also coexist with added structure of non-algebraic nature, such as partial order or a topology. The added structure must be compatible, in some sense, with the algebraic structure.
Universal algebra.
Algebraic structures are defined through different configurations of axioms. Universal algebra 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 variety (not to be confused with algebraic varieties 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 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 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". The axioms can be represented as trees. These equations induce equivalence classes on the free algebra; the quotient algebra then has the algebraic structure of a group.
Some structures do not form varieties, because either:
Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g., fields and division rings. Structures with nonidentities present challenges varieties do not. For example, the direct product of two fields is not a field, because formula_1, but fields do not have zero divisors.
Category theory.
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 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
Different meanings of "structure".
In a slight abuse of notation, 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 formula_2," means that we have defined ring "operations" on the set formula_2. For another example, the group formula_4 can be seen as a set formula_5 that is equipped with an "algebraic structure," namely the "operation" formula_6.