While trying to unify the (mostly linearly) ordered attributes from Statistical classification with the binary attributes from formal concept analysis, I came across an interesting section in the wikipedia article on the Dedekind–MacNeille completion:
If $O$ is any finite set of objects, and $A$ is any finite set of binary attributes for the objects in $O$, then one may form a partial order of height two in which the elements of the partial order are the objects and attributes, and in which $x \leq y$ when $x$ is an object that has attribute $y$. For a partial order defined in this way, the Dedekind–MacNeille completion of $S$ is known as a concept lattice, and it plays a central role in the field of formal concept analysis.
While this seems to point to an easy way to use the Dedekind–MacNeille completion to achieve the desired unification, the statement made in this section is not completely true1. However, essentially the same statement can also be found in the wikipedia article on Formal concept analysis:
It may be viewed as the Dedekind–MacNeille completion of a partially ordered set of height two in which the elements of the partial order are the objects and attributes of M and in which two elements x and y satisfy x ≤ y exactly when x is an object that has attribute y.
Question: Where does this statement come from, and how must it be modified to express a "really correct" relation between concept lattice and Dedekind–MacNeille completion?
1. The completion will contain each single object (and each single attribute), but the concept lattice only contains a concept for an object, if its attributes differ from all other objects. In other words, in the concept lattice, similar objects and similar attributes are merged together into a single concept, but the Dedekind–MacNeille completion won't merge anything.