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up vote 1 down vote favorite 1 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? Is 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},m