In
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 ...
, alternativity is a property of 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 ...
. 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 ...
is said to be if
for all
and if
for all
. A magma that is both left and right alternative is said to be ().
[.]
Any
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 ...
magma (that is, a
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The
converse, however, is not true, in contrast to the situation in
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.
Examples
Examples of alternative algebras include:
* Any
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
is associative and therefore alternative.
*
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 alternative and flexible but not associative. See for more examples.
*
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or ...
multiplication is alternative and flexible.
** More generally
Cayley-Dickson algebra over a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
is alternative.
See also
*
Flexible algebra
In mathematics, particularly abstract algebra, a binary operation • 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 ...
*
Power associativity
References
Properties of binary operations
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