Q:

Complete Lattice and Concept Lattice

I am taking an online course of Introduction to Formal Concept Analysis and I'm trying to understand The Basic Theorem.
Well, if $\mathscr{L}$ is a complete lattice, so $\mathscr{L}\cong \underline{\mathfrak{B}}(\mathscr{L},\mathscr{L},\leq)$ (concept lattice of context $(\mathscr{L},\mathscr{L},\leq)$).
And we know that:
(*) All concept lattice are complete lattice.
My question is: When we define a formal context $(G,M,I)$, the sets $G,M$ don't have 'conditions'. What happens when I take a non-complete lattice $\mathscr{L}$ and define the formal context $(\mathscr{L},\mathscr{L},\leq)$? Can I do this? If yes, so $\underline{\mathfrak{B}}(\mathscr{L},\mathscr{L},\leq)$ is not $\mathscr{L}$?
For instance. Take a non-complete lattice $(\mathbb{N},\leq$) and the formal context $(\mathbb{N},\mathbb{N},\leq)$. And about the concept lattice of this context?
Thanks for a help!

A:

Formal concepts can be identified with certain subsets of $G$ (specifically with the closed subsets in the Galois connection induced by $I$), which are closed under taking arbitrary intersections. 
Note that it is happening in the already complete lattice of the power set.
The case $(\mathcal L, \mathcal L, \le)$ with $\mathcal L$ a complete lattice is special. Then, for subsets $A, B\subseteq\mathcal L$ we have
$$\forall a\in A,\,b\in B\, (a\le b) \ \iff\ \bigvee A\le\bigwedge B$$
so that the subsets can be represented by single elements. 

