Terms and notation for the sets of upper and lower bounds of a given set.

Question

I've seen this notation for the upper and the lower sets generated by a subset $X$ in a poset $P$: $$ P\uparrow X =\{\,y\in P\mid(\exists x\in X)(y\ge x)\,\},\qquad P\downarrow X =\{\,y\in P\mid(\exists x\in X)(y\le x)\,\}. $$

I am looking for terms and notation for the sets $$ \{\,y\in P\mid(\forall x\in X)(y\ge x)\,\} \quad\text{and}\quad \{\,y\in P\mid(\forall x\in X)(y\le x)\,\}. $$ Is there any?

I would also be interested in the notation for the sets of strict upper and lower bounds $$ \{\,y\in P\mid(\forall x\in X)(y > x)\,\} \quad\text{and}\quad \{\,y\in P\mid(\forall x\in X)(y < x)\,\}. $$

Answer 1

In the context of the Dedekind-MacNeille completion of a partially ordered set I have seen the notation $X^u$ and $X^\ell$ for the sets of upper and lower bounds of $X$.