Infimum and supremum And Net

Question

We know that every Partially Ordered Set has to satisfy three conditions :

1. Reflexive

2. Anti-Symmetric

3. Transitive

If we have the partially ordered set $S$ with a relation $R$, and $S$ also satisfies the following two conditions: For any two elements $a$ and $b$ such that $a \subseteq S$ and $b \subseteq S$:

1. there is always $m \subseteq S$ where $m$ is equal to $\inf\,\{a,b\}$; (the infimum of $a$ and $b$),

2. and also there will be $n \subseteq S$ where $n$ is equal to $\sup\,\{a,b\}$; (the supremum of $a$ and $b$)

In this case, does the set $S$ has a special name? Is it possible to define a net on this set? I know my question is so silly but I am beginner in this field.

I also want to ask if we have a net $N$ has been defined on a partially ordered set $S$ that satisfies the previous conditions (1, 2) AND also: Every subset of $S$ also satisfies the same conditions (1, 2), in this case, what can we call The net $N$? Any help will mean a lot.