Understanding the Basic Theorem on Concept Lattices In Ganter and Wille's Applied Lattice Theory: Formal Concept Analysis, one can find the following definition:

Basic Theorem on Concept Lattices. Let $K := (G, M, I)$ be a formal context. Then $\mathcal B(K)$ is a complete lattice, called the concept lattice of $(G, M, I)$, for which infimum and supremum can be described as follows:
  $\land_{t\in T}(A_t,B_t)=\left(\bigcap_{t\in T}A_t,\left(\bigcup_{t\in T}B_t\right)''\right)$,  $\lor_{t\in T}(A_t,B_t)=\left(\left(\bigcup_{t\in T}A_t\right)'', \bigcap_{t\in T}B_t\right)$.

where $A \subseteq G$ and $B \subseteq M$. It is based on Restructuring Lattice Theory: An Approach Based on Hierarchies of Concepts.
As far as I understand, the second formula (supremum) refers to the topmost node in a lattice. But what about the first formula (infimum)? Is it the lowermost node? If yes, shouldn't there be exists instead of all at the beginning? It sounds to me like all formal concepts in a complete lattice must have the form of the lowermost node, which doesn't make any sense to me. How should it be interpreted correctly?
 A: There are no quantifiers in these two formulas. I suppose you mixed them ($\forall$ and $\exists$) up with the lattice meet $\wedge$ and join $\vee$ operations. 
The complete lattice is a poset in which all subsets have both a supremum and an infimum. So you have two operations $\bigvee$ and $\bigwedge$ (these are just the notations for $\mbox{sup}$ and $\mbox{inf}$ and in case of complete lattices the big signs are usually used instead of small $\vee$ and $\wedge$).
As you probably know the formal concept $C = (A, B)$ consists of the extent $A$ and the intent $B$. The theorem states just that the set of all concepts considered as a poset with the ordering $C_1 \leqslant C_2$ if and only if $\mbox{ext}(C_1) \subseteq \mbox{ext}(C_2)$ forms a complete lattice and the formulas above just provide you the way to compute the infimum and the supremum in this lattice. For clarity, you may replace $\wedge$ with $\mbox{inf}$ and $\vee$ with $\mbox{sup}$ obtaining:
$$\mbox{inf} \{(A_t, B_t) \mid t \in T\} = \left(\bigcap_{t\in T}A_t,\left(\bigcup_{t\in T}B_t\right)''\right)\\
\mbox{sup} \{(A_t, B_t) \mid t \in T\} = \left(\left(\bigcup_{t\in T}A_t\right)'', \bigcap_{t\in T}B_t\right).$$
Informally, in terms of formal concepts you may consider the following interpretation. Let's say that $C_1$ is more general than $C_2$ if $\mbox{int}(C_1) \subseteq \mbox{int}(C_2)$. That means that $C_1$ has less features and therefore can be used to describe more objects, that's why it is "more general". In this case, $C_2$ is called more special than $C_1$ for the same reasons - it has more features, so it can be used to describe only some specific set of objects.  


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*The infimum $\bigwedge K$ of $K = \{(A_t, B_t) \mid t \in T\}$ is the "direct specialization" of concepts.

*The supremum $\bigvee K$ of $K = \{(A_t, B_t) \mid t \in T\}$ is the "direct generalization" of concepts.
