Wolfram Cloud Page

Which of these algebraic relations have standard names : {a·b  b·a, a  (a·b)·(a·a), (a·a)·(a·a)  a, (a·b)·(b·b)  b, a  (a·b)·(a·(b·c)), a  (a·b)·(a·(c·b)), a  (a·b)·((b·c)·a), a  (a·b)·((c·b)·a), a  (b·a)·(a·(b·c)), a  (b·a)·((b·c)·a)}

Let's analyze these algebraic relations to see which ones have standard names:1. Commutative Property:

•

is known as the Commutative Property of multiplication.

For the remaining relations, they do not correspond to well-known properties in standard algebra or abstract algebra. They appear to be custom or specific algebraic identities that might be used in specialized algebraic structures or contexts, but they do not have standard names like the commutative property.If you need further exploration into any specific algebraic structure or context where these identities might appear, feel free to ask!

What about the others?

Let's examine the remaining relations to see if they correspond to any known algebraic properties or structures:2. Idempotent Law:

•

does not correspond to a standard algebraic property. However, the idempotent law usually states

, which is somewhat related but not identical.

3. Idempotent Property:

•

is a variation of the idempotent law, but it's not a standard form. In some algebraic structures, an element is idempotent if

.

4. Absorption Law:

•

doesn't match a standard property in common algebraic systems. The absorption laws typically involve combinations of meet and join operations in lattice theory.

5. Associative-like Variations:

•

These resemble associative-like transformations but don't correspond to standard associative properties. Associativity typically refers to rearranging parentheses without changing the order of elements, i.e.,

.6. Non-standard Variations:

•

These don't correspond to well-known properties and appear to be specific forms that might hold in certain algebraic systems or under specific conditions but aren't standard names.Overall, aside from the commutative property identified initially, the other relations are not standard in typical algebraic contexts. They could be part of specialized systems or identities used in particular mathematical theories. If you have a specific context or system in mind, that could provide more insight into these relations.

Which of these algebraic relations have standard names : {a·b  b·a, a  (a·b)·(a·a), (a·a)·(a·a)  a, (a·b)·(b·b)  b, a  (a·b)·(a·(b·c)), a  (a·b)·(a·(c·b)), a  (a·b)·((b·c)·a), a  (a·b)·((c·b)·a), a  (b·a)·(a·(b·c)), a  (b·a)·((b·c)·a)}

Certainly! Let's analyze each of the given algebraic relations and identify which ones have standard names in algebra.

1. Commutative Property:

• Relation:

• Standard Name: This is known as the Commutative Law of multiplication.

• Explanation: It states that the order in which two elements are multiplied does not affect the result.

2. Non-standard Relation:

• Relation:

• Standard Name: This does not correspond to a standard named identity in common algebraic structures.

• Explanation: While it involves elements and operations, it doesn't match any well-known property.

3. Power Associativity / Alternative Property:

• Relation:

• Standard Name: This is related to the concept of Power Associativity and properties in Alternative Algebras, but it's not a standard identity with a specific name.

• Explanation: It suggests that repeated multiplication of an element leads back to the original element, a property seen in certain algebraic structures.

4. Non-standard Relation:

• Relation:

• Standard Name: This relation does not have a recognized standard name.

• Explanation: It appears specific and doesn't correspond directly to standard algebraic properties.

5. Moufang Identity:

• Relation:

• Standard Name: This is a form of the Moufang Identity.

• Explanation: Moufang identities are important in the study of Moufang Loops, which are non-associative algebraic structures. This particular form is one of the equivalent conditions defining a Moufang loop.

6. Moufang Identity (Variation):

• Relation:

• Standard Name: Another variation of the Moufang Identity.

• Explanation: This identity is another equivalent form used in Moufang loops.

7. Moufang Identity (Variation):

• Relation:

• Standard Name: Yet another form of the Moufang Identity.

• Explanation: This identity shows a different arrangement but is still a Moufang identity.

8. Moufang Identity (Variation):

• Relation:

• Standard Name: Also a variation of the Moufang Identity.

• Explanation: It is an alternative form seen in the context of Moufang loops.

9. Moufang Identity (Variation):

• Relation:

• Standard Name: Another variation of the Moufang Identity.

• Explanation: This rearranged form continues to represent the properties of Moufang loops.

10. Moufang Identity (Variation):

• Relation:

• Standard Name: Variant of the Moufang Identity.

• Explanation: This is one of the multiple forms that the Moufang identity can take.

Summary of Standard Names:

• Relation 1: Commutative Law

• Relations 5–10: Moufang Identities

Additional Information on Moufang Identities:The Moufang Identities are a set of equivalent identities that characterize Moufang Loops, which are non-associative algebraic structures generalizing groups. These identities help compensate for the lack of associativity by introducing alternative ways elements can be combined.Key Moufang Identities:1. First Moufang Identity:

•

2. Second Moufang Identity:

•

3. Third Moufang Identity:

•

These identities are crucial in the theory of Moufang loops and have applications in geometry and the study of non-associative algebras, such as octonions.

In Conclusion:

• The commutative law is well-known in many algebraic structures where the order of multiplication does not affect the outcome.

• The Moufang identities are recognized standard identities in the study of non-associative structures like Moufang loops.

• The other relations provided do not have standard names in algebra and are specific identities that may be applicable in particular contexts or specialized algebraic systems.

Feel free to ask if you need further clarification on any of these identities or their applications!