Understanding Identity Elements in Abstract Binary Operations

[dijkarte]

When we talk about an abstract binary operation *:SxS --> S

We say that an identity element "e" exists when e * x = x * e = x, (x; e belong to S)

Now my questions:

1) If the operation is not commutative, does not this imply no identity? since e * x != x * e necessarily?

2) Does e * x = x imply x * e = x? And if one side is true, can we still say that * has an identity restricted to one side?

3) Is there cases where not all x's of the set S have the same identity wrt operation *?

These questions are regardless of any kind of algebraic structure.

Thanks.

 

[micromass]

When we talk about an abstract binary operation *:SxS --> S

We say that an identity element "e" exists when e * x = x * e = x, (x; e belong to S)

Now my questions:

1) If the operation is not commutative, does not this imply no identity? since e * x != x * e necessarily?

We DEMAND that e*x=x*e. If it is not true, then e can not be an identity.
If the operation is commutative, then asking that e*x=x*e is unnecessary. So it is only interesting for noncommutative operations.

2) Does e * x = x imply x * e = x? And if one side is true, can we still say that * has an identity restricted to one side?

When e is an identity, we DEMAND that both e*x=x and x*e=x are true. If it is not true, then e is not an identity.
However, if the structure is commutative, then one equality implies the other. So demanding that both equalities hold is only interesting for noncommutative operations.

3) Is there cases where not all x's of the set S have the same identity wrt operation *?

No. An identity must be one for ALL elements. That is, for ALL x it must be that e*x=x*e=x.
If it is only true for one x, then e is not an identity.

 

[dijkarte]

Thank you very much. This clarified things to me.