It seems to me that if I have strict partial ordering relation such as $<$, I can always derive a partial ordering relation $\leq$ simply by defining $\leq$ to mean $<$ or $=$. And vice versa, given a partial ordering relation $\leq$ I can always define a strict partial ordering relation by defining $<$ as $\leq$ and not $=$.
What is the most general case for which this is true? Is it true for every partial ordering relation? For example, do there exist ordering relations in cases where no equivalence relation exists?
Is there terminology to describe the inclusion/exclusion of the equality condition to/from a given relation? How can I evoke this in a way which will be clear to the reader? For example can I say the following: if $R$ is a partial ordering relation, let $R'$ be its corresponding strict partial ordering.
Can I say the following? If $R$ and $R'$ are corresponding relations then, $R$ is reflexive, symmetric, and transitive, if and only if $R'$ is irreflexive and transitive.
I hope what I'm trying to express is clear. Can someone help me say it cleanly?
Indeed every strict partial order goes along with a partial order and vice versa.
Let $X$ be a set let $\Delta:=\{\langle x,x\rangle\mid x\in X\}$.
It is true in this context that:$$R\text{ is a partial order on }X\implies R-\Delta\text{ is a strict partial order on }X$$ and: $$S\text{ is a strict partial order on }X\implies S\cup\Delta\text{ is a partial order on }X$$
So partial order $R$ induces strict partial order by $S:=R-\Delta$.
On its turn strict partial order $S$ induces partial order $T:=S\cup\Delta$.
Then it is easy to verify that $T=R$ and the circle is closed.