In
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 term ''a ...
, the sedenions form a 16-
dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordi ...
noncommutative
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 ...
and
nonassociative algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s; they are obtained by applying 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 b ...
to the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s, and as such the octonions are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to a
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operat ...
of the sedenions. Unlike the octonions, the sedenions are not an
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 ...
. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the ''32-ions'' or ''trigintaduonions''. It is possible to continue applying the Cayley–Dickson construction arbitrarily many times.
The term ''sedenion'' is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the
biquaternion
In abstract algebra, the biquaternions are the numbers , where , and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions co ...
s, or the algebra of 4 × 4
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
over the real numbers, or that studied by .
Arithmetic
![Sedenion-Fano Tesseract](https://upload.wikimedia.org/wikipedia/commons/4/44/Sedenion-Fano_Tesseract.gif)
Like
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s,
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 additi ...
of sedenions is neither
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 ...
nor
associative.
But in contrast to the octonions, the sedenions do not even have the property of being
alternative
Alternative or alternate may refer to:
Arts, entertainment and media
* Alternative (''Kamen Rider''), a character in the Japanese TV series ''Kamen Rider Ryuki''
* ''The Alternative'' (film), a 1978 Australian television film
* ''The Alternative ...
.
They do, however, have the property of
power associativity In mathematics, specifically in abstract algebra, power associativity is a property of a binary operation that is a weak form of associativity.
Definition
An algebra (or more generally a magma) is said to be power-associative if the subalgebra ge ...
, which can be stated as that, for any element
x of
, the power
is well defined. They are also
flexible
Flexible may refer to:
Science and technology
* Power cord, a flexible electrical cable.
** Flexible cable, an Electrical cable as used on electrical appliances
* Flexible electronics
* Flexible response
* Flexible-fuel vehicle
* Flexible rake re ...
.
Every sedenion is a
linear combination of the unit sedenions
,
,
,
, ...,
,
which form a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of the
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
of sedenions. Every sedenion can be represented in the form
:
Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is
distributive over addition.
Like other algebras based on 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 b ...
, the sedenions contain the algebra they were constructed from. So, they contain the octonions (generated by
to
in the table below), and therefore also the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s (generated by
to
),
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s (generated by
and
) and real numbers (generated by
).
The sedenions have a multiplicative
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
and multiplicative inverses, but they are not a
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
because they have
zero divisors
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
. This means that two non-zero sedenions can be multiplied to obtain zero: an example is
. All
hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.
A sedenion multiplication table is shown below:
Sedenion properties
From the above table, we can see that:
:
:
and
:
Anti-associative
The sedenions are not fully anti-associative. Choose any four generators,
and
. The following 5-cycle shows that these five relations cannot all be anti-associative.
In particular, in the table above, using
and
the last expression associates.
Quaternionic subalgebras
The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have e ...
s used in creating the sedenion through 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 b ...
shown in bold:
The binary representations of the indices of these triples
bitwise XOR
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operati ...
to 0.
The list of 84 sets of zero divisors
, where
:
Applications
showed that the space of pairs of norm-one sedenions that multiply to zero is
homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
to the compact form of the exceptional
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
G2. (Note that in his paper, a "zero divisor" means a ''pair'' of elements that multiply to zero.)
Sedenion neural networks provide a means of efficient and compact expression in machine learning applications and have been used in solving multiple time-series and traffic forecasting problems.
See also
*
Hypercomplex number
In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group represent ...
*
Pfister's sixteen-square identity
*
Split-complex number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
Notes
References
*
*
*
*
*
*
*
*L. S. Saoud and H. Al-Marzouqi, "Metacognitive Sedenion-Valued Neural Network and its Learning Algorithm," in IEEE Access, vol. 8, pp. 144823-144838, 2020, doi
10.1109/ACCESS.2020.3014690
{{Number systems
Hypercomplex numbers
Non-associative algebras