[SOUND] [MUSIC] Concept lattices provide lossless representation of data. If you have a properly labelled concept lattice, you can reconstruct its formal context. This is often useful, but not always. Consider the following example. It has 12 attributes. And maybe these attributes are courses taught at some faculty at a university. And objects of this context are students. Many of the students take the same courses, so they're represented by the same line in our data table. So we have 100 students that take courses a, b, c; 100 students that take courses d, e, f; 70 students take courses g, h, i; and 30 students take courses j, k, l. Well, maybe this faculty has four educational programs and these educational programs are completely disjoint. They don't share any courses. Then we get this formal context, and this is its concept lattice. It has a very simple structure. Now let's see what happens if we get one exchange student. So an exchange student, just one exchange student, comes to this university and she decides to take a course a, d, g, and j; so she takes one course from each of the programs. Well, exchange students can do this. So what happens to the lattice now? Well, we still have the same concepts, the same six concepts, but we get five more. So this is the labelling. We have attribute a here, attribute d here, attribute g here, and attribute j here. And, here we have, bc, ef, hi, and kl. And so our original concepts are, well, this one, the top one, the bottom one, abc, def, ghi, and jkl. But because of this one exchange student, we got five more concepts. So the extent of this concept contains just one student, this one. And here we have a concept with the intent {a}, which groups all students who took the course a. And this is all students from the first educational program, the 100 students that took a, b, and c, and this exchange student; and so on for the other extra concepts. Well, now if you forget about the numbers, and you look at the structure of these two lattices, then, well, these are pretty different lattices. And still, they describe roughly the same situation. The only difference is that here we have this one additional extra person, an exchange student. And maybe you want to see this information, because somehow it tells you that there's something in common between courses from different programs, a, d, g, and j. But if all you want is to study the general structure of the education at this faculty, then you really wouldn't like to see these five extra concepts that are due to the exchange student that comes from elsewhere. And if you add more exchange students, so, let's say, another five exchange students come, and they take different sets of courses, you can imagine that the concept lattice can become really huge. And so from its structure it will be pretty difficult to understand how the education is really organized at this faculty. So it would be nice to be able to identify these extra concepts and somehow remove them from the lattice. And today we're going to discuss a couple of techniques that allow us to do this. [SOUND] [MUSIC]