Consider a set $S$ which is a lattice and denote the join operation $\vee$.
Should I think of $x\vee y$ as $x\sup y$ or as $x\sup_S y$ (where $\sup_s$ means sup with respect to $S$)
Here is an illustration of my confusion. Let $$S=\big\{ (0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)\big\}$$ and consider $$T=S\setminus\{(1,1), (1,2),(2,1)\} = \big\{ (0,0),(0,1),(0,2),(1,0),(2,0),(2,2)\big\}$$
Both $S$ and $T$ are lattices but is $(0,1)\wedge (1,0) = (1,1)$ or $(2,2)?$ The sup of these points in $T$ is $(2,2)$ but in $S$ it is $(1,1)$
I ask because for $T$ to be a sublattice of $S$ i believe $T$ needs to be closed under $\vee\ ,\ \wedge$. But if I define these operations w.r.t $T$ then it is a sub lattice and if i define them w.r.t to $S$ then it is not.
For elements $x, y$ in a lattice, $x\lor y$ is the same as their supremum, i.e. the lowest common upper bound of $x$ and $y$ within the lattice.
So that, $(1,0)\lor(0,1)$ is $(1,1)$ in $S$ and is $(2,2)$ in $T$ (note that $T$ doesn't even contain $(1,1)$).
It also means that $T$ is not a sublattice of $S$, just a 'subposet' which happen to have lowest common upper bounds.