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For a set to be a monoid, it must be associative and must have an identity element. I've proved that is it associative but don't know how to prove that it has a identity element.

  1. Let $\mathbb{R}^3$ be the set of all ordered triples of numbers, and let $\otimes$ be the binary operation on $\mathbb{R}^3$ defined such that $$\left(x_1, y_1, z_1\right) \otimes \left(x_2, y_2, z_2\right) = \left(x_1x_2, x_1y_2+y_1z_2, z_1z_2\right)$$ for all $\left(x_1, y_1, z_1\right), \left(x_2, y_2, z_2\right) \in \mathbb{R}^3$. Prove that $\left(\mathbb{R}^3, \otimes\right)$ is a monoid. What is the identity element of this monoid? Is the monoid $\left(\mathbb{R}^3, \otimes\right)$ a group?
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    $\begingroup$ It is not hard to see from the expression that the identity element must be of the form $(1,y,1)$. For what value of $y=y_1$ does one have $1y_2+yz_2=y_2$ for all $(x_2,y_2,z_2)$? Does one also have $(x_1,x_2,x_3)\otimes(1,y,1)=(x_1,x_2,x_3)$ for this value of $y$ and all $(x_1,x_2,x_3)$? $\endgroup$ – Marc van Leeuwen Jan 23 '13 at 16:58
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Suppose that there exist a tuple $(x_2,y_2,z_2)$ such that $(x_1,y_1,z_1)\otimes (x_2,y_2,z_2)=(x_1,y_1,z_1)$ for all $(x_1,y_1,z_1)\in \mathbb{R}^3$. Looking at the first coordinate of your formula we get $x_1x_2=x_1$ for all $x_1$ and this implies that $x_2=1$. In the same way, but looking at the third coordinate, we get $z_1z_2=z_1$ for all $z_1$, so that $z_2=1$. Now, it is easy get $y_2=0$. Since the law it is not commutative you need to check the other way around. This is an easy computation and it shows that $(1,0,1)$ is indeed the identity.

This monoid is not a group. For example the element $(0,0,0)$ does not have an inverse because if $(x,y,z)$ were such inverse then $(0,0,0)\otimes (x,y,z)$ would be $(1,0,1)$ but $(0,0,0)\otimes (x,y,z)=(0,0,0)$.

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  • $\begingroup$ So does that also mean it's a group? $\endgroup$ – Adegoke A Jan 23 '13 at 17:07
  • $\begingroup$ To see that it is not a group you can observe that $(0,0,0)\otimes(x,y,z)=(0,0,0)$ for all $(x,y,z)$ so that $(0,0,0)$ does not have inverse. $\endgroup$ – Quimey Jan 23 '13 at 17:10
  • $\begingroup$ Is there a reason for using (0,0,0)? Why not invert (1,0,1)? $\endgroup$ – Adegoke A Jan 23 '13 at 17:13
  • $\begingroup$ In a group every element is invertible. The element $(0,0,0)$ is not invertible in your monoid. In the other hand $(1,0,1)$ is his own inverse (and therefore it is invertible$. $\endgroup$ – Quimey Jan 23 '13 at 17:20
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    $\begingroup$ No, It is not a group. I have edited the answer to include these observations. $\endgroup$ – Quimey Jan 23 '13 at 17:25
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Looks suspiciously like the set of matrices \begin{equation} \begin{bmatrix} x & y\\ 0 & z \end{bmatrix} \end{equation} under matrix multiplication.

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I would try something like $(1,0,1)$ and $(1,1,1)$. In first and third coordinates the operation is just mulitplication, so the unit will be $1$ there.

But the clearest if you write it up:

$(x,y,z)\otimes (1,b,1) = (x,y,z)\ $ and solve it for $b$. You can use specific $(x,y,z)$'s, e.g. plug in $(1,0,0)$, etc.

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Well, you need to find $(x_2,y_2,z_2)$ s.t. $x_1x_2=x_1$, $x_1y_2+y_1z_2=y_1$ and $z_1z_2=z_1$. From this it follows that $x_2=z_2=1$. And $y_2=0$.

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So for $(e,f,g)$ to be an identity element means in particular that $$(x,y,z)\otimes (e,f,g)=(x,y,z)$$ for all $x,y,z$. Comparing coordinates one gets $xe=x$, $xf+yg=y$ and $zg=z$.

Solving this and checking this and checking which elements satisfy the reverse identity is left as an exercise for you.

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  • $\begingroup$ How did you get $xf + yg = y$ ? $\endgroup$ – Adegoke A Apr 10 '13 at 23:43
  • $\begingroup$ @AdegokeA This is just comparison of the second coordinate of the result of this "strange" multiplication and $(x,y,z)$. $\endgroup$ – Julian Kuelshammer Apr 11 '13 at 6:36
  • $\begingroup$ But should it not be $xe = x, yf =y$ and $zg = z$ ? $\endgroup$ – Adegoke A Apr 11 '13 at 10:32
  • $\begingroup$ @AdegokeA No, see the definition of $\otimes$ in the question. $\endgroup$ – Julian Kuelshammer Apr 11 '13 at 10:54
  • $\begingroup$ It says a binary operation. I thought ⊗ meant multiplication. Those the ⊗ operator have a different function? $\endgroup$ – Adegoke A Apr 11 '13 at 11:43
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The identity element is (1,0,1)

Also $(x,y,z)^{-1}=(x^{-1},\frac{-y}{zx},z^{-1})$, if $x\not=0,z\not=0$

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    $\begingroup$ This formula makes no sense for $x=0$ (or $z$). $\endgroup$ – Quimey Jan 23 '13 at 17:01
  • $\begingroup$ Is it really a group? $\endgroup$ – Adegoke A Jan 23 '13 at 17:07
  • $\begingroup$ If you get rid of the zeros in x and z, then yes. $\endgroup$ – Ishan Banerjee Jan 23 '13 at 17:09

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