Turning preorders into partial orders Given a preorder $\preceq$ we can define a partial order $\leq$ as:


*

*$x<y$ iff $x\preceq y$ and not $y\preceq x$

*$x\leq y$ iff $x<y$ or $x=y$


Transitivity is inherited from $\preceq$, antisymmetry comes from (1) and reflexivity from (2).
My question: is there a name for this? If I just refer to "the partial order constructed from $\preceq$" will people understand what I mean?

I would call the partial order constructed in this was the 'antisymmetric restriction' of the preorder. I imagine others might call it the maximal antisymmetric subrelation. I don't believe there's a standard notation or name for such a construction.
