Algebraic structure

In mathematics, more specifically in abstract algebra and universal algebra, 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. Some algebraic structures also involve another set (called the scalar set).

In the context of universal algebra, the set A with this structure is called an algebra,^[1] 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.

Examples of algebraic structures with a single underlying set include groups, rings, fields, and lattices. Examples of algebraic structures with two underlying sets include vector spaces, modules, and algebras.

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[edit]

Addition and multiplication on 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 law. Also a + b = b + a and ab = ba as 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 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 number of sets and operations that can combine more than two elements (higher arity), 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 Category:Algebraic structures. Structures are listed in approximate order of increasing complexity.

Examples[edit]

One set with operations[edit]

Simple structures: no binary operation:

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

Magma or groupoid: S and a single binary operation over S.
Semigroup: an associative magma.
Monoid: a semigroup with identity element.
Group: a monoid with a unary operation (inverse), giving rise to inverse elements.
Abelian group: a group whose binary operation is commutative.
Semilattice: a semigroup whose operation is idempotent and commutative. The binary operation can be called either meet or join.
Quasigroup: a magma obeying the Latin square property. A quasigroup may also be represented using three binary operations.^[2]

Loop: a quasigroup with identity.

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

Semiring: 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 absorbing element in the sense that 0 x = 0 for all x.
Near-ring: a semiring whose additive monoid is a (not necessarily abelian) group.
Ring: a semiring whose additive monoid is an abelian group.
Lie ring: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the Jacobi identity rather than associativity.
Commutative ring: a ring in which the multiplication operation is commutative.
Boolean ring: a commutative ring with idempotent multiplication operation.
Field: a commutative ring which contains a multiplicative inverse for every nonzero element.
Kleene algebras: a semiring with idempotent addition and a unary operation, the Kleene star, satisfying additional properties.
*-algebra: a ring with an additional unary operation (*) satisfying additional properties.

Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law.^[3]

Complete lattice: a lattice in which arbitrary meet and joins exist.
Bounded lattice: a lattice with a greatest element and least element.
Complemented lattice: a bounded lattice with a unary operation, complementation, denoted by postfix ^⊥. The join of an element with its complement is the greatest element, and the meet of the two elements is the least element.
Modular lattice: a lattice whose elements satisfy the additional modular identity.
Distributive lattice: a lattice in which each of meet and join distributes over the other. Distributive lattices are modular, but the converse does not hold.
Boolean algebra: 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.
Heyting algebra: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix →, and governed by the axioms x → x = 1, x (x → y) = x y, y (x → y) = y, x → (y z) = (x → y) (x → z).

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.

Robinson arithmetic. Addition and multiplication are recursively 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.
Peano arithmetic. Robinson arithmetic with an axiom schema of induction. 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[edit]

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

Group with operators: a group G with a set Ω and a binary operation Ω × G → G satisfying certain axioms.
Module: an abelian group M and a ring R acting as operators on M. The members of R are sometimes called scalars, and the binary operation of scalar multiplication is a function R × M → M, which satisfies several axioms. Counting the ring operations these systems have at least three operations.
Vector space: a module where the ring R is a division ring or field.
Graded vector space: a vector space with a direct sum decomposition breaking the space into "grades".
Quadratic space: 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).

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.

Algebra over a ring (also R-algebra): a module over a commutative ring R, which also carries a multiplication operation that is compatible with the module structure. This includes distributivity over addition and linearity with respect to multiplication by elements of R. The theory of an algebra over a field is especially well developed.
Associative algebra: an algebra over a ring such that the multiplication is associative.
Nonassociative algebra: 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 alternation, the Jacobi identity, or the Jordan identity.
Coalgebra: a vector space with a "comultiplication" defined dually to that of associative algebras.
Lie algebra: a special type of nonassociative algebra whose product satisfies the Jacobi identity.
Lie coalgebra: a vector space with a "comultiplication" defined dually to that of Lie algebras.
Graded algebra: 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.
Inner product space: an F vector space V with a sesquilinear binary operation V × V → F.

Four or more binary operations:

Bialgebra: an associative algebra with a compatible coalgebra structure.
Lie bialgebra: a Lie algebra with a compatible bialgebra structure.
Hopf algebra: a bialgebra with a connection axiom (antipode).
Clifford algebra: a graded associative algebra equipped with an exterior product from which may be derived several possible inner products. Exterior algebras and geometric algebras are special cases of this construction.

Hybrid structures[edit]

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.

Topological group: a group with a topology compatible with the group operation.
Lie group: a topological group with a compatible smooth manifold structure.
Ordered groups, ordered rings and ordered fields: each type of structure with a compatible partial order.
Archimedean group: a linearly ordered group for which the Archimedean property holds.
Topological vector space: a vector space whose M has a compatible topology.
Normed vector space: a vector space with a compatible norm. If such a space is complete (as a metric space) then it is called a Banach space.
Hilbert space: an inner product space over the real or complex numbers whose inner product gives rise to a Banach space structure.
Vertex operator algebra
Von Neumann algebra: a *-algebra of operators on a Hilbert space equipped with the weak operator topology.

Universal algebra[edit]

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:

It is necessary that 0 ≠ 1, 0 being the additive identity element and 1 being a multiplicative identity element, but this is a nonidentity;
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., 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 $(1,0)\cdot (0,1)=(0,0)$ , but fields do not have zero divisors.

Category theory[edit]

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"[edit]

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 $A$ ," means that we have defined ring 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 $+$ .

Notes[edit]

^ P.M. Cohn. (1981) Universal Algebra, Springer, p. 41.
^ Jonathan D. H. Smith. An Introduction to Quasigroups and Their Representations. Chapman & Hall. Retrieved 2012-08-02.
^ 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.

References[edit]

Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (2nd ed.), AMS Chelsea, ISBN 978-0-8218-1646-2
Michel, Anthony N.; Herget, Charles J. (1993), Applied Algebra and Functional Analysis, New York: Dover Publications, ISBN 978-0-486-67598-5
Burris, Stanley N.; Sankappanavar, H. P. (1981), A Course in Universal Algebra, Berlin, New York: Springer-Verlag, ISBN 978-3-540-90578-3

Category theory

Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2
Taylor, Paul (1999), Practical foundations of mathematics, Cambridge University Press, ISBN 978-0-521-63107-5

External links[edit]

Jipsen's algebra structures. Includes many structures not mentioned here.
Mathworld page on abstract algebra.
Stanford Encyclopedia of Philosophy: Algebra by Vaughan Pratt.

[1] P.M. Cohn. (1981) Universal Algebra, Springer, p. 41.

[2] Jonathan D. H. Smith. An Introduction to Quasigroups and Their Representations. Chapman & Hall. Retrieved 2012-08-02.

[3] 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.

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Tuesday, September 1, 2020