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If $xy$ is a unit, are $x$ and $y$ units?
There is no doubt if a and b are units then ab is a unit. How about the converse? Still holds?
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No. See this previous question. Though it is about rings, forgetting the addition will give you the examples for monoids. – Arturo Magidin Jan 30 '12 at 01:00
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What do you mean by "unit"? Usually it means the identity element, but there can only be one of those in a monoid, so I'll assume you mean an invertible element.
If the monoid is commutative, then $ab$ invertible easily implies $a$ invertible and $b$ invertible.
In a non-commutative monoid: if $ab$ is right invertible, then $a$ is also right invertible, but we cannot say anything about $b$ in particular. (For example, in the monoid generated by $a$, $b$ and $c$ with the single relation $abc=1$, $ab$ is right invertible, but $b$ is neither left nor right invertible).
hmakholm left over Monica
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@FortuonPaendrag: Yes, in a ring. But it must have surprised me to see the terminology used in a non-ring setting. – hmakholm left over Monica Aug 05 '12 at 12:28
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This is untrue in general, but $a$ and $b$ do have one-sided inverses.
Proof: If for $a,b \in M$, $ab$ is a unit, then $\exists k \in M$ such that $(ab)k=k(ab)=1$.Since multiplication in a momoid is associative, we have that $a(bk)=1$ and $(ka)b=1$, demonstrating explicitly a right inverse for $a$ and a left inverse for $b$. But as Arturo Magidin's example-which he has linked to-demonstrates, this may be the best we can obtain.
Arkady
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1You've only demonstrated that $a$ has a right inverse and that $b$ has a left inverse. Could you please show why $(bk)a=1$ and $b(ka)=1$ as well? Before you try, be sure to consider the example here. – Arturo Magidin Jan 30 '12 at 01:11
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@ArturoMagidin Ah, I just saw your comment and was reading through it. Thank you. I will edit accordingly! – Arkady Jan 30 '12 at 01:13
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@ArturoMagidin If we know av=bv implies a=b, then this converse holds. v(kuv)=v implies (vku)v=v, which means vku=1, that is v(ku)=1. – Shannon Jan 30 '12 at 02:02
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@Shannon: Sure, extra hypothesis may suffice, but it's not true in general. – Arturo Magidin Jan 30 '12 at 02:35