Flexible algebra

A flexible algebra is a non-associative algebra over a field ( -algebra), for the multiplication of which the law of flexibility applies. ${\ displaystyle K}$

Examples

• Every Lie algebra is a flexible algebra because its multiplication is the Lie bracket (see above).
• The multiplication of octonions and sedions fulfills the law of flexibility.
• An alternativegebra (i.e. a non-associative algebra whose multiplication is alternative ) is a flexible algebra. ${\ displaystyle K}$
• Here, the flexibility of the multiplication follows from the alternative together with the -bilinearity of the multiplication (the representation of the link symbol for the multiplication is omitted in the following): ${\ displaystyle K}$
• For true because of Linksalternativität of multiplication: ${\ displaystyle a + b}$
• ${\ displaystyle ((a + b) (a + b)) b = (a + b) ((a + b) b)}$
• repeated application of the -bilinearity of the multiplication ("multiply") results ${\ displaystyle K}$
• ${\ displaystyle \ Leftrightarrow (aa) b + (ab) b + (ba) b + (bb) b = a (ab) + a (bb) + b (ab) + b (bb)}$
• ${\ displaystyle \ Leftrightarrow (ba) b = (a (ab) - (aa) b) + (a (bb) - (ab) b) + b (ab) + (b (bb) - (bb) b) }$
• the first and third differences disappear because of the left alternativity of the multiplication, the second difference disappears because of the right alternativity of the multiplication. So it follows:
• ${\ displaystyle \ Leftrightarrow (ba) b = b (ab)}$

Flexibility Act

In mathematics, the law of flexibility is understood to be the following rule for a link ${\ displaystyle \ circ}$

${\ displaystyle a \ circ \ left (b \ circ a \ right) = \ left (a \ circ b \ right) \ circ a}$.

The law of flexibility is automatically fulfilled by commutative or associative operations :

• From (associativity) follows with directly .${\ displaystyle a \ circ \ left (b \ circ c \ right) = \ left (a \ circ b \ right) \ circ c}$${\ displaystyle c = a}$${\ displaystyle a \ circ \ left (b \ circ a \ right) = \ left (a \ circ b \ right) \ circ a}$
• With the doubly applied commutative law applies . ${\ displaystyle a \ circ \ left (b \ circ a \ right) \; {\ stackrel {1.} {=}} \, \ left (b \ circ a \ right) \ circ a \; {\ stackrel { 2.} {=}} \, \ Left (a \ circ b \ right) \ circ a}$
1. because of with${\ displaystyle a \ circ x = x \ circ a}$${\ displaystyle x = b \ circ a}$
2. because of ${\ displaystyle b \ circ a = a \ circ b \ Rightarrow \ left (b \ circ a \ right) \ circ a = \ left (a \ circ b \ right) \ circ a}$

The law of flexibility becomes significant when a link is no longer associative and no longer commutative and thus still allows "bracketing" within a modest framework.

Alternative links usually do not comply with the Flexibility Act, see counterexample below .

Examples

Due to their antisymmetry and linearity, the following applies:
${\ displaystyle \ left [a, [b, a] \ right] = - [[b, a], a] = [- [b, a], a] = [[a, b], a]}$.
• The multiplication of an alternative field fulfills the flexibility law. Here the flexibility of multiplication follows from its alternative, the group axioms of addition and the laws of distributors. The proof is analogous to the proof of the flexibility of the multiplication in an alternativegebra using the distributive laws instead of bilinearity.

Flexible magmas

A magma that is linked to the law of flexibility is also called a flexible magma .

Examples

• Each semigroup is a flexible magma, since the associative law gives rise to the flexibility law (see above).
• The magma with the following linkage table is flexible but not alternative :
${\ displaystyle \ circ}$ 0 1
0 1 0
1 0 0
• not alternatively because of ${\ displaystyle 0 \ circ (0 \ circ 1) = 1 \ neq 0 = (0 \ circ 0) \ circ 1}$
• flexible because of
• ${\ displaystyle 0 \ circ (0 \ circ 0) = 0 = (0 \ circ 0) \ circ 0}$
• ${\ displaystyle 0 \ circ (1 \ circ 0) = 1 = (0 \ circ 1) \ circ 0}$
• ${\ displaystyle 1 \ circ (0 \ circ 1) = 0 = (1 \ circ 0) \ circ 1}$
• ${\ displaystyle 1 \ circ (1 \ circ 1) = 0 = (1 \ circ 1) \ circ 1}$
• The magma with the following linkage table is alternative but not flexible :
${\ displaystyle \ circ}$ 0 1 2
0 0 0 2
1 0 0 2
2 0 1 2
• not flexible because of ${\ displaystyle 1 \ circ (2 \ circ 1) = 0 \ neq 1 = (1 \ circ 2) \ circ 1}$
• left alternative because of
• ${\ displaystyle 0 \ circ (0 \ circ 0) = 0 = (0 \ circ 0) \ circ 0}$
• ${\ displaystyle 0 \ circ (0 \ circ 1) = 0 = (0 \ circ 0) \ circ 1}$
• ${\ displaystyle 0 \ circ (0 \ circ 2) = 2 = (0 \ circ 0) \ circ 2}$
• ${\ displaystyle 1 \ circ (1 \ circ 0) = 0 = (1 \ circ 1) \ circ 0}$
• ${\ displaystyle 1 \ circ (1 \ circ 1) = 0 = (1 \ circ 1) \ circ 1}$
• ${\ displaystyle 1 \ circ (1 \ circ 2) = 2 = (1 \ circ 1) \ circ 2}$
• ${\ displaystyle 2 \ circ (2 \ circ 0) = 0 = (2 \ circ 2) \ circ 0}$
• ${\ displaystyle 2 \ circ (2 \ circ 1) = 1 = (2 \ circ 2) \ circ 1}$
• ${\ displaystyle 2 \ circ (2 \ circ 2) = 2 = (2 \ circ 2) \ circ 2}$
• alternative legal because of
• ${\ displaystyle 0 \ circ (0 \ circ 0) = 0 = (0 \ circ 0) \ circ 0}$
• ${\ displaystyle 0 \ circ (1 \ circ 1) = 0 = (0 \ circ 1) \ circ 1}$
• ${\ displaystyle 0 \ circ (2 \ circ 2) = 2 = (0 \ circ 2) \ circ 2}$
• ${\ displaystyle 1 \ circ (0 \ circ 0) = 0 = (1 \ circ 0) \ circ 0}$
• ${\ displaystyle 1 \ circ (1 \ circ 1) = 0 = (1 \ circ 1) \ circ 1}$
• ${\ displaystyle 1 \ circ (2 \ circ 2) = 2 = (1 \ circ 2) \ circ 2}$
• ${\ displaystyle 2 \ circ (0 \ circ 0) = 0 = (2 \ circ 0) \ circ 0}$
• ${\ displaystyle 2 \ circ (1 \ circ 1) = 0 = (2 \ circ 1) \ circ 1}$
• ${\ displaystyle 2 \ circ (2 \ circ 2) = 2 = (2 \ circ 2) \ circ 2}$
• There is no alternative magma with fewer than three elements that is not flexible.
• The magma with the following linkage table is alternative and flexible , but not associative :
${\ displaystyle \ circ}$ 0 1 2
0 0 0 2
1 0 1 1
2 0 1 2
• not associative because of ${\ displaystyle 0 \ circ (1 \ circ 2) = 0 \ neq 2 = (0 \ circ 1) \ circ 2}$
• flexible because of (without link symbol)
• ${\ displaystyle 0 (00) = 0 = (00) 0}$
• ${\ displaystyle 0 (10) = 0 = (01) 0}$
• ${\ displaystyle 0 (20) = 0 = (02) 0}$
• ${\ displaystyle 1 (01) = 0 = (10) 1}$
• ${\ displaystyle 1 (11) = 1 = (11) 1}$
• ${\ displaystyle 1 (21) = 1 = (12) 1}$
• ${\ displaystyle 2 (02) = 2 = (20) 2}$
• ${\ displaystyle 2 (12) = 1 = (21) 2}$
• ${\ displaystyle 2 (22) = 2 = (22) 2}$
• left alternative because of (without link symbol)
• ${\ displaystyle 0 (00) = 0 = (00) 0}$
• ${\ displaystyle 0 (01) = 0 = (00) 1}$
• ${\ displaystyle 0 (02) = 2 = (00) 2}$
• ${\ displaystyle 1 (10) = 0 = (11) 0}$
• ${\ displaystyle 1 (11) = 1 = (11) 1}$
• ${\ displaystyle 1 (12) = 1 = (11) 2}$
• ${\ displaystyle 2 (20) = 0 = (22) 0}$
• ${\ displaystyle 2 (21) = 1 = (22) 1}$
• ${\ displaystyle 2 (22) = 2 = (22) 2}$
• right alternative because of (without link symbol)
• ${\ displaystyle 0 (00) = 0 = (00) 0}$
• ${\ displaystyle 0 (11) = 0 = (01) 1}$
• ${\ displaystyle 0 (22) = 2 = (02) 2}$
• ${\ displaystyle 1 (00) = 0 = (10) 0}$
• ${\ displaystyle 1 (11) = 1 = (11) 1}$
• ${\ displaystyle 1 (22) = 1 = (12) 2}$
• ${\ displaystyle 2 (00) = 0 = (20) 0}$
• ${\ displaystyle 2 (11) = 1 = (21) 1}$
• ${\ displaystyle 2 (22) = 2 = (22) 2}$
• There is no alternative and flexible magma with fewer than three elements that is not associative.