A lattice is defined if a poset $T$ has a infimum and supremum for any finite $S \subseteq T$.
What is a lattice called, where $\forall S \in \mathcal{P}(T). inf(S) \in S \wedge sup(S) \in S$?
I have a feeling that such a lattice would always be isomorphic to $\mathbb{Z}$.
A lattice $L$ for which every subset $A$ we have $\bigvee A \in A$ and $\bigwedge A \in A$ is necessarily a finite chain, that is, its ordered elements look like $$a_1 \prec a_2 \prec \cdots \prec a_n,$$ where $a \prec b$ means $a < b$ and there exist no $c$ with $a < c < b$.
In this case, $\bigwedge A = \min A$ and $\bigvee A = \max A$.
Indeed, if there exist $a,b \in L$ with $a \nleq b$ and $b \nleq a$, then set $A = \{a,b\}$ and you have that $\bigwedge A \notin A$ and $\bigvee A \notin A$.
If there is an infinite chain $(a_n)_{n \in \mathbb N}$ and $a_n < a_{n+1}$, then set $A = \{a_n: n \in \mathbb N\}$, and $\bigvee A \notin A$.
If there is an infinite chain $(a_n)_{n \in \mathbb N}$ and $a_n > a_{n+1}$, then set $A = \{a_n: n \in \mathbb N\}$, and $\bigwedge A \notin A$.
So you see that actually $\mathbb Z$ is not one such lattice...