Trigeno/Q/Almost ordering/0 maximum/Exercise

On the recently discovered planet Trigeno there lives a species which has some ability to calculate. They use like us the rational numbers with "our“ addition and multiplication. They also use a kind of "ordering“ on the rational numbers, denoted by ${}\succeq$. This trigenometric ordering coincides with our ordering as long as both numbers are${}\neq 0$. However, they put

${}0\succeq x\,$

for every rational number ${}x$. The well-known Ethnomathematician Dr. Eisenbeis thinks that this is related to the fact that they worship the number ${}0$.

Show that ${}\succeq$ fulfils the following properties.

1. For any two elements ${}a,b\in \mathbb {Q}$ either ${}a\succ b$ or ${}a=b$ or ${}b\succ a$ holds.
2. From ${}a\succeq b$ and ${}b\succeq c$ we get ${}a\succeq c$ (for arbitrary ${}a,b,c\in \mathbb {Q}$).
3. From ${}a\succeq 0$ and ${}b\succeq 0$ we get ${}a+b\succeq 0$.
4. From ${}a\succeq 0$ and ${}b\succeq 0$ we get ${}ab\succeq 0$.

Which property of an ordered field does ${}(\mathbb {Q} ,\succeq )$ not fulfil?

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