Right inverse, left inverse, binary operations

[Panphobia]

Homework Statement

If * is a binary operation on a set B, and the domain of definition is B^2, if * is associative and the neutral element is p. If r and l are elements of b we can say that r is a left inverse of l under * iff r * l = p, and l is a right inverse of r iff l * r = p. Show that if an element of B has a left and right inverse, then they are equal.

The Attempt at a Solution

Does the neutral element have anything to do with finding the answer, also what does associativity have to do with finding the answer? All I can think of is since there is only one neutral element in *, r = p and l = p, but I don't think that is the answer.

 

[Dick]

Homework Statement

If * is a binary operation on a set B, and the domain of definition is B^2, if * is associative and the neutral element is p. If r and l are elements of b we can say that r is a left inverse of l under * iff r * l = p, and l is a right inverse of r iff l * = p. Show that if an element of B has a left and right inverse, then they are equal.

The Attempt at a Solution

Does the neutral element have anything to do with finding the answer, also what does associativity have to do with finding the answer? All I can think of is since there is only one neutral element in *, r = p and l = p, but I don't think that is the answer.

Sure, you have to assume there is a neutral element p, so that for any x, x*p=p*x=x. Now suppose r is a right inverse of x (so x*r=p) and l is a left inverse of x (so l*x=p). Associativity tells you (l*x)*r=l*(x*r), yes? So?

 

[Panphobia]

Does it matter than the question say l * r = p and r * l = p, where does the x come from? would this mean (l * r) * l = l * (r * l)?

 

[Dick]

Does it matter than the question say l * r = p and r * l = p, where does the x come from? would this mean (l * r) * l = l * (r * l)?

The letters were confusing and I changed them. Your writing (l * r) * l = l * (r * l) shows why. The definitions you've got are using 'l' to be a symbol for both the right and left inverse. They are two different things. Those definitions mean something independent of the particular symbols used. Suppose I tell you that b is a right inverse of c. Let's keep the symbol 'p' to mean the neutral element, though I probably would have used the symbol '1' instead. It's more suggestive. What does that mean?

 

[Panphobia]

Then why in the question is l * r = p, and r * l = p defined? btw r and l do not mean right and left inverse, they are arbitrary variables, the question says, "l is left inverse of r under * iff l * r = p, and l is a right inverse of r under * iff r * l = p". It is kind of confusing.

 

[D H]

First off, look at some numbers you know, such as the rationals. With * being the multiplication operator you learned long ago and p being 1, what is the left inverse of 2 ? The right inverse? The answer is obviously 1/2 in both cases: 1/2*2 = 2*1/2 = 1. The goal of this problem is to show that if an element of B has both left and right inverses and if * is associative then there is only one inverse (i.e., the left and right inverses are one and the same).

Note that this result does not necessarily mean that * is commutative. What it does mean is that an element of B commutes with its multiplicative inverse.

Dick took the correct approach by giving that element of B that has both a left and right inverse a new symbol, x.

 

[Dick]

Then why in the question is l * r = p, and r * l = p defined? btw r and l do not mean right and left inverse, they are arbitrary variables, the question says, "l is left inverse of r under * iff l * r = p, and l is a right inverse of r under * iff r * l = p". It is kind of confusing.

Yes, they are arbitrary variables, and yes, it's kind of confusing. The reason it's confusing is because they made such a poor choice of arbitrary symbols. Try to unscramble it. You didn't answer my question. I'll repeat it. Suppose I tell you that b is a right inverse of c. What does that mean? Bypass the whole 'l' and 'r' thing.

 

[Panphobia]

c * b = p, where p is the neutral element

 

[Dick]

c * b = p, where p is the neutral element

Bingo! Now I'll tell you that d is a left inverse of c. Same question. And then can you show b=d?

 

[Panphobia]

(d * c) * b = d * (c * b)
d * c = c * b = p
p * b = d * p
so this implies d = b, right?

 

[Dick]

(d * c) * b = d * (c * b)
d * c = c * b = p
p * b = d * p
so this implies d = b, right?

Right. Couldn't be clearer. I'm almost wondering if they didn't choose the symbol choice in original definitions to try to test you and throw you off track.

 

[Panphobia]

Thanks for the help!