Alternative implies powers up to the fifth are well-defined
This article gives the statement and possibly, proof, of an implication relation between two magma properties. That is, it states that every magma satisfying the first magma property (i.e., alternative magma) must also satisfy the second magma property (i.e., magma in which powers up to the fifth are well-defined)
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Contents
Statement
Suppose is an alternative magma, i.e., it satisfies the following two identities:
and
These are respectively termed the left-alternative law and right-alternative law.
Then cubes, fourth powers, and fifth powers are well-defined in . In other words, if denotes , we have the following for all :
- Cubes are well-defined: , and this is denoted as .
- Fourth powers are well-defined: , and this is denoted as .
- Fifth powers are well-defined: and this is denoted as .
Related facts
- Jordan implies powers up to the fifth are well-defined
- Left alternative and flexible implies powers up to the fifth are well-defined
- Right alternative and flexible implies powers up to the fifth are well-defined
Proof
As above, we assume is an alternative magma and is an arbitrary element of .
Cubes are well-defined
To prove: .
Proof: This follows from either of the alternative laws, setting .
Fourth powers are well-defined
To prove: .
Proof: We first show that . Consider:
By the right-alternative law, setting , we obtain that:
We now show that . We have:
By the left-alternative law, setting , we obtain:
.
Thus, we have shown that all three expressions are equal.
Fifth powers are well-defined
To prove: .
Proof:
We first show that . Note that:
where, in an intermediate step, we used the right-alternative law with .
We next show that . Note that:
where, in an intermediate step, we used the right-alternative law with .
We finally show that . Note that:
where, in an intermediate step, we used the left-alternative law with .
This completes the proof.