In mathematics, particularly

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

• on a set is flexible if it satisfies the flexible identity: :

a \bullet \left(b \bullet a\right) = \left(a \bullet b\right) \bullet a

for any two elements ''a'' and ''b'' of the set. A

magma Magma () is the molten or semi-molten natural material from which all igneous rocks are formed. Magma is found beneath the surface of the Earth, and evidence of magmatism has also been discovered on other terrestrial planets and some natura ...

(that is, a set equipped with a binary operation) is flexible if the binary operation with which it is equipped is flexible. Similarly, a

nonassociative algebra A non-associative algebra (or distributive algebra) is an algebra over a field where the binary multiplication operation is not assumed to be associative. That is, an algebraic structure ''A'' is a non-associative algebra over a field ''K'' if ...

is flexible if its multiplication operator is flexible. Every

commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...

associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...

operation is flexible, so flexibility becomes important for binary operations that are neither commutative nor associative, e.g. for the

multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being ad ...

sedenion In abstract algebra, the sedenions form a 16- dimensional noncommutative and nonassociative algebra over the real numbers; they are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphi ...

s, which are not even alternative. In 1954, Richard D. Schafer examined the algebras generated by the Cayley–Dickson process over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

and showed that they satisfy the flexible identity.Richard D. Schafer (1954) “On the algebras formed by the Cayley-Dickson process”,

American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United ...

76: 435–46

Examples

Besides associative algebras, the following classes of nonassociative algebras are flexible: *

Alternative algebra In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have *x(xy) = (xx)y *(yx)x = y(xx) for all ''x'' and ''y'' in the algebra. Every associative algebra is ...

s *

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...

s *

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan al ...

s (which are commutative) * Okubo algebras Similarly, the following classes of nonassociative magmas are flexible: *

Alternative magma In abstract algebra, alternativity is a property of a binary operation. A magma ''G'' is said to be if (xx)y = x(xy) for all x, y \in G and if y(xx) = (yx)x for all x, y \in G. A magma that is both left and right alternative is said to be ( ...

s *

Semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...

s (which are associative magmas, and which are also alternative) The

s, and all algebras constructed from these by iterating the

Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, named after Arthur Cayley and Leonard Eugene Dickson, produces a sequence of algebras over the field of real numbers, each with twice the dimension of the previous one. The algebras produced by ...

, are also flexible.

References

* {{cite book , first=Richard D. , last=Schafer , year=1995 , origyear=1966 , zbl=0145.25601 , title=An introduction to non-associative algebras , publisher=

Dover Publications Dover Publications, also known as Dover Books, is an American book publisher founded in 1941 by Hayward and Blanche Cirker. It primarily reissues books that are out of print from their original publishers. These are often, but not always, books ...

, isbn=0-486-68813-5 , url-access=registration , url=https://archive.org/details/introductiontono0000scha Non-associative algebras Properties of binary operations

Examples

See also

References