If you have a language $\mathcal L$ with a binary relation $R$ and function $f(x,y)$, you can write formulae such as $\forall x\exists y(R(f(x,y),x)\iff R(y,x))$.
When interpreting the language in a certain structure $M$ we use $R^M$ to denote the relation and $f^M$ to denote the function.
This is just an example. Suppose your language has $\cup$ and $\cap$ for functions. If you have a lattice $L$ and you want to say that this is the lattice in which you interpret your functions in you can write $\cap^L$ and $\cup^L$.
If you want to write a general statement which holds for a lattice with certain properties, you can write a sentence $\varphi$. If $L$ is a lattice in which the statement is true, then you can write $L\models\varphi$, or "$\varphi$ holds in $L$", that is to say that taking your world to be $L$ with the functions $\cap^L$ and $\cup^L$ the sentence is true.
Suppose you want to write a theorem for lattices with a certain property then you can say: "If $L$ is a lattice with such and such properties, then $\varphi$ holds in $L$".
I suppose that this is what you ultimately want to do.