mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, particularly

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...

, 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, a binary operation ...

• on a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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 (sometimes colloquially but incorrectly referred to as ''lava'') is found beneath the surface of the Earth, and evidence of magmatism has also ...

(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 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. Perhaps most familiar as a pr ...

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

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

multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...

sedenion In abstract algebra, the sedenions form a 16-dimension of a vector space, dimensional commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers, usually represented by the cap ...

s, which are not even

alternative Alternative or alternate may refer to: Arts, entertainment and media * Alternative (Kamen Rider), Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki'' * Alternative comics, or independent comics are an altern ...

. 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 S ...

76: 435–46

Examples

Besides

associative algebra In mathematics, an associative algebra ''A'' over a commutative ring (often a field) ''K'' is a ring ''A'' together with a ring homomorphism from ''K'' into the center of ''A''. This is thus an algebraic structure with an addition, a mult ...

s, the following classes of nonassociative algebras are flexible: *

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

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 ident ...

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 a ...

s (which are commutative) * Okubo algebras In the world of magmas, there is only a binary multiplication operation with no addition or scaling with from a base ring or field like in algebras. In this setting,

and

magmas are all flexible - the alternative and commutative laws all imply flexibility. This includes many important classes of magmas: all groups, semigroups and

moufang loop In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by . Smooth Moufang loops have an associated algebra, the Malcev algebra, ...

s are flexible. The

s and

trigintaduonion In abstract algebra, the trigintaduonions, also known as the , , form a commutative property, noncommutative and associative property, nonassociative algebra over a field, algebra over the real numbers. Names The word ''trigintaduonion'' is d ...

s, and all algebras constructed from these by iterating the

Cayley–Dickson construction In mathematics, the Cayley–Dickson construction, sometimes also known as the Cayley–Dickson process or the Cayley–Dickson procedure produces a sequence of algebra over a field, algebras over the field (mathematics), field of real numbers, eac ...

, 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, book ...

, 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