Is a finite monoid with left cancellation property always a group?

Question

I need to answer and show if a Monoid with left cancellation property always a group.

I managed to show that it is correct when cancellation property holds for both left and right (that was part a of the question), but I have a feeling i proved it wrong because i didn't use the right side at all. what I did is :

M is finite though if $x \in M$ there is some $i \in N$ such that

$x = x^i$

then

$x*e = x*x^{i-1}$

so

$x^{i-1} = e$

then $x^{i-2}$ is the inverse of x

so i proved that G is a Group without using the right cancelation property.

where am I wrong..

any help will be appreciated

Your very first assumption is wrong. In a finite monoid you don't need $x=x^i$ for some $i$ ($i$ may not exist). — Crostul, Sep 01 '15 at 19:23
what do you mean i may not exists? how come? if it is finite , we can't have $ x, x^2 , x^3..... $ we have to have an i that $x^i = e $ . no? — user2993422, Sep 01 '15 at 19:29
If you don't have cancellation, you can only say that $x^i = x^j$ for some $i \not= j$, but you cannot force $i$ to be $1$. — J.-E. Pin, Sep 01 '15 at 19:32
why? if $x^i = x^j$ and i < j , then $x^{j-i} = e$ then $x^{j-i+1} = x$ am I wrong? — user2993422, Sep 01 '15 at 19:37

Dietrich Burde · Accepted Answer · 2015-09-01T19:38:29.807

2

Yes, every finite monoid $M$ with left (or right) cancellation is a group, because the mapping $x\mapsto ax$ is injective, and since $M$ is finite, it is already bijective by the pigeonhole (box) principle. Hence $M$ is a group.
Your argument above is not correct as demonstrated in the comments, but it can be corrected by using the cancellation property.

edited Sep 01 '15 at 19:38

answered Sep 01 '15 at 19:32

Dietrich Burde

140,055

The OP asks where is the mistake in his argument. – Crostul Sep 01 '15 at 19:35
@Crostul You have explained this mistake already. Finally one needs some "box" argument and cancellation. – Dietrich Burde Sep 01 '15 at 19:40
@DietrichBurde, very nice answer! – user237522 Nov 27 '24 at 16:36

Is a finite monoid with left cancellation property always a group?

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