Potency Associative Algebra

A power-associative algebra is an algebra in which the powers of an element can be defined independently of the order of brackets.

Definitions

Define for one magma and each ${\displaystyle {\mathcal {M}}=(M,\circ )}$ ${\displaystyle a\in M}$

${\displaystyle a^{1}:=a}$as well as for each . ${\displaystyle a^{k+1}:=a\circ (a^{k})}$ ${\displaystyle k\in \mathbb {N} }$

The linking of a magma 's potency associative for an item if for all positive integers applies ${\displaystyle \circ }$ ${\displaystyle (M,\circ )}$ ${\displaystyle a\in M}$ ${\displaystyle i,j\in \mathbb {N} ^{*}}$

${\displaystyle a^{i+j}=(a^{i})\circ (a^{j})}$

A magma is called a potency-associative magma if its connection is potency-associative for each . ${\displaystyle {\mathcal {M}}=(M,\circ )}$ ${\displaystyle \circ }$ ${\displaystyle a\in M}$

Algebra is called potency-associative ( potency-associative algebra ) if its multiplication is potency-associative, i.e. is a potency-associative magma. ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \cdot }$ ${\displaystyle ({\mathcal {A}},\cdot )}$

Examples

Potency-Associative Magmas

• Each semigroup is also always a potency-associative magma.
• For every idempotent the potency associativity is true element of a magma: . Correspondingly, every idempotent magma is a potency-associative magma ${\displaystyle a^{i+j}=a=a\circ a=(a^{i})\circ (a^{j})}$
  
• Every alternative and flexible connection that fulfills the Moufang identities is also potency-associative.
  Proof (by complete induction ):
  • Induction start : ${\displaystyle i=1}$ ${\displaystyle a^{1}\circ a^{j}=^{1}a\circ a^{j}=^{1}a^{j+1}=a^{1+j}}$
  • Induction start : ${\displaystyle i=2}$ ${\displaystyle a^{2}\circ a^{j}=^{1}(a\circ a)\circ a^{j}=^{2}a\circ (a\circ a^{j})=^{1}a\circ a^{1+j}=^{1}a^{2+j}}$
  • Induction step for : ${\displaystyle i\longrightarrow i+1}$ ${\displaystyle i\geq 2}$
    ${\displaystyle a^{i+1}\circ a^{j}=^{1}(a\circ a^{i})\circ a^{j}=^{1}(a\circ (a\circ a^{i-1}))\circ a^{j}}$
    ${\displaystyle =^{3}(a\circ (a^{i-1}\circ a))\circ a^{j}}$
    ${\displaystyle =^{4}a\circ (a^{i-1}\circ (a\circ a^{j}))}$
    ${\displaystyle =^{1}a\circ (a^{i-1}\circ a^{j+1})}$
    ${\displaystyle =^{5}a\circ a^{(i-1)+(j+1)}=a\circ a^{i+j}=^{1}a^{(i+j)+1}=a^{(i+1)+j}}$

1: definition ${\displaystyle a^{n}}$

2: (Left) alternative of ${\displaystyle \circ }$

3: Flexibility (and the resulting i-power associativity, see below) of ${\displaystyle \circ }$

4: Moufang identity for ${\displaystyle \circ }$

5: Induction assumption

• For the multiplication of an algebra, the alternative is sufficient, see below!
• For special cases, fewer requirements are sufficient. So it follows from the alternative: 1: Definition 2: Left alternativity 3: Right alternativity ${\displaystyle a^{3+2}=a^{3}a^{2}}$
  ${\displaystyle a^{3+2}=a^{5}=^{1}a(a(a(aa)))=^{2}a((aa)(aa))=^{3}(a(aa))(aa)=^{1}a^{3}a^{2}}$
  ${\displaystyle a^{n}}$
  
  

Potency-Associative Algebras

• All associative algebras are power-associative.

• All alternative algebras are power-associative.
  • In an algebra, the flexibility of the multiplication follows from the alternative, and also the fulfillment of the Moufang identities (see also properties of alternative fields )!

• All -algebras in which, for every one is having are potency-associative. ${\displaystyle K}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle a\in {\mathcal {A}}}$ ${\displaystyle c_{a}\in K}$ ${\displaystyle a\cdot a=c_{a}\cdot a}$
  • This includes, for example , equipped with the cross product, there for everyone . ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle a\times a=0}$ ${\displaystyle a\in \mathbb {R} ^{3}}$
• The algebra of the sedions is also a potency-associative algebra.

Further weakening of power associativity

The linking of a magma 's i-potency-associative for an item if for any positive integer applies ${\displaystyle \circ }$ ${\displaystyle (M,\circ )}$ ${\displaystyle a\in M}$ ${\displaystyle i\in \mathbb {N} ^{*}}$

${\displaystyle a^{i}\circ a=a\circ a^{i}}$

A magma, the connection of which is i-power-associative, can therefore also be referred to as an i-power-associative magma .

A potency-associative magma is also always an i-potency-associative magma, because the following applies:

${\displaystyle a\circ (a^{i})=^{1}a^{i+1}=^{2}(a^{i})\circ (a^{1})=^{1}a^{i}\circ a}$

1: definition ${\displaystyle a^{n}}$

2: power associativity of ${\displaystyle \circ }$

A flexible magma (and especially every semigroup) is always an i-potency-associative magma, because it applies (by complete induction ):

• Induction start (only with definition ): ${\displaystyle i=1}$ ${\displaystyle a^{n}}$ ${\displaystyle a^{1}\circ a=a\circ a=a\circ a^{1}}$
• Induction step : ${\displaystyle i\longrightarrow i+1}$ ${\displaystyle a^{i+1}\circ a=^{1}(a\circ a^{i})\circ a=^{2}a\circ (a^{i}\circ a)=^{3}a\circ (a\circ a^{i})=^{1}a\circ a^{i+1}}$

1: definition ${\displaystyle a^{n}}$

2: flexibility of ${\displaystyle \circ }$

3: Induction assumption

The connection of a magma is called idemassociative (based on idempotent ) for an element , if applies ${\displaystyle \circ }$ ${\displaystyle (M,\circ )}$ ${\displaystyle a\in M}$

${\displaystyle a\circ (a\circ a)=(a\circ a)\circ a}$.

A magma whose connection is idemassociative can therefore also be referred to as an idemassociative magma .

An i-potency-associative magma is also always an idemassociative magma (with ). ${\displaystyle i=2}$

Examples

1. The magma with the following link table is idem-associative, but neither i-potency-associative (and thus also not potency-associative) nor flexible nor alternative:

${\displaystyle \circ }$	0	1	2
0	2	1	2
1	2	2	0
2	2	0	0

• not left alternative because of ${\displaystyle 0\circ (0\circ 1)=0\circ 1=1\neq 0=2\circ 1=(0\circ 0)\circ 1}$
• not legal alternative because of ${\displaystyle 0\circ (2\circ 2)=0\circ 0=2\neq 0=2\circ 2=(0\circ 2)\circ 2}$
• not flexible because of ${\displaystyle 1\circ (0\circ 1)=2\neq 0=(1\circ 0)\circ 1}$
• not potency-associative because of ${\displaystyle 0^{2+2}=0^{4}=0\circ (0\circ (0\circ 0)=2\neq 0=(0\circ 0)\circ (0\circ 0)=0^{2}\circ 0^{2}}$
• not i-potency-associative because of ${\displaystyle 1\circ 1^{3}=1\circ (1\circ (1\circ 1))=2\neq 1=(1\circ (1\circ 1))\circ 1=1^{3}\circ 1}$
• idem-associative because of
  • ${\displaystyle 0\circ (0\circ 0)=2=(0\circ 0)\circ 0}$
  • ${\displaystyle 1\circ (1\circ 1)=0=(1\circ 1)\circ 1}$
  • ${\displaystyle 2\circ (2\circ 2)=2=(2\circ 2)\circ 2}$

2. The magma with the following link table is potency-associative (and thus also i-potency-associative and idem-associative), but neither flexible nor alternative:

${\displaystyle \circ }$	0	1	2
0	0	0	0
1	0	0	2
2	0	0	2

• not alternatively because of ${\displaystyle 1\circ (1\circ 2)=2\neq 0=(1\circ 1)\circ 2}$
• not flexible because of ${\displaystyle 2\circ (1\circ 2)=2\neq 0=(2\circ 1)\circ 2}$
• potency-associative because of
  • ${\displaystyle 0^{i+j}=0=0\circ 0=0^{i}\circ 0^{j}}$
  • ${\displaystyle 1^{i+j}=0=0\circ 0=1^{i}\circ 1^{j}}$
  • ${\displaystyle 2^{i+j}=2=2\circ 2=2^{i}\circ 2^{j}}$

3. The exponentiation is not idemassoziativ (and therefore neither I nor potency associative potency-associative), as it applies, for example: . ${\displaystyle \left(3^{3}\right)^{3}=27^{3}=19683\neq 7625597484987=3^{27}=3^{\left(3^{3}\right)}}$

literature

• Ebbinghaus et al .: Numbers . Springer, Berlin 1992, ISBN 3-54055-654-0 .
• RD Schafer: An Introduction to Nonassociative Algebras . Benediction Classics, 2010, ISBN 1-84902-590-8 .