are all partial orders subsets of some lattice?

Question

Take a partially ordered set, $(X,\succ)$. Does there exist a Lattice $(Y,\succ')$, such that $X\subset Y$ and $\forall x,x'\in X$, $$x\succ x'\iff x\succ'x'$$

A more formal statement of this question: For any partially ordered set, $(X,\succ)$, does there exist a Lattice $(Y,\succ')$ and a function $f:X\rightarrow Y$ that is order embedding?

I apologize if this question may have an obvious reference, but such a reference would be much appreciated.

Answer 1

Yes, analogously to Dedekind cuts there is a way to "complete" a partial order to form a lattice.