Constructing an order-preserving function between two strict p.o.s that is not one-to-one.

Question

Out of Winfried Just and Martin Weese's Set theory book:

Give an example of strict partial orders $\langle X,<_x\rangle$ and $\langle Y,<_y\rangle$ and of a function $\tau : X \to Y$ that is not one-to-one, and yet has the property that $\forall x,y\in X\big(x<_xy\iff\tau(x)<_y\tau(y)\big)$.

But if $\tau(x)$ is not one-to-one there exists $\tau(y)$ such that $\tau(x)=\tau(y)$ and since $<_y$ is a strict partial order, $(\tau(x),\tau(y))\notin<_y$ since $<_y$ is irreflexive. I must be misunderstanding something since this would imply there is no such function.

The next question has the additional restriction that the strict partial orders are linear orders and asks to prove that $\tau(x)$ must be one-to-one. However I don't see how imposing linear ordering changes my contradiction above. Thoughts?

Answer 1

HINT:

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