[MUSIC] Can we compute with concepts? Well, in a way, we can. If we had a concept of dogs and we have, on the other hand, a concept of cats. When we want to compute the concept that generalizes them both, like for example, we could say that the concept of all animals generalizes both the concept of a dog and the concept of a cat, because every dog is an animal and every cat is an animal. The concept of domestic animal, or a pet, also generalizes dogs and cats, but it's more specific than the concept of animal. Or on the other hand if we have a concept of triangles and a concept of all geometric figures that have a right angle, we may want to compute a concept that includes those that fall on the both these concepts. So the concept this in a way, it specializes both these concepts, it would include right angled triangles. Now let's see how we can do things like this with formal concepts. So, again, let's say that we have two formal concepts of the same formal concept (G,M,I). And the concepts are (A1,B1) and (A2,B2). We say that the greatest common subconcept, Or we'll also call it infimum, Of these three concepts. Let's denote it like this, A1, B1 A2, B2, so A1, B1 wedge A2, B2 is by definition the fallen formal concept. Its extent is the intersection of A1 and A2. And its intent is the closure of the union of B1 and B2. So, the intuition is like this, we want to define the greatest common subconcept of two concepts, A1, B1, and A2, B2. So objects that fall under this concept must fall under both A1, B1 and A2, B2 that's why we take the intersection of A1 and as an example of this new concept and these objects. They all have properties from both B1 and B2. That's why in the intent of the new formal concept we have B1 union B2, but because intents must be closed, we take the closure. And similarly we can define the least general, super concept. We'll also call it supremum of A1, B1. And A2, B2. In this case we will take the closure of the union of A1 and A2 as the extent and the intersection of the intent. So B1 into section B2 as the intent of the new concept. What we want from these, Subconcepts and superconcepts, actually, we want four things. So let's look at this one, A1, B1 wedge A2, B2. This is a greatest common of concepts. So the first thing we want from it is that it is actually a concept. So A1, B1 wedge A2, B2 must be a formal concept of the context GMI. And it is, I'll skip the proof, it's not very difficult, but a little bit boring and technical. However, it's not sufficient for this to be a concept, it must be a subconcept of both (A1, B1) and (A2, B2), it must be their common subconcept. So we should also have that A1, That this infimum is indeed a subconcept of A1, B1. And it's also a subconcept of A2, B2. And this is easy to see, it's extend is A1 intersection A2, which is a sub set of both A1 and A2. So by definition This a subconcept of both, A1 B1 and A2 B2. However, we need not only any common subconcept, but the greatest common subconcept. In other words, if there is any other subconcept, the common subconcept of A1,B1 and A2,B2. It must be, Less general than this one. So for any concept (A,B) Such that (A,B) is a subconcept of (A1,B1) and it's also a sub concept of A2B2. So for every such concept (A,B), we have that. (A,B) is also a subconcept of the infimum. Again, it takes some time to prove this, but this can be proven. So, the fourth point says that the infimum is indeed the greatest commons of concept of the two concepts and the same, or similar four points, hold for the supremum. Now, to get some feeling for all this supremum and infimum, let's look at an example. I'm going to draw a concept lattice and we'll see how you can find the infimum and supremum of formal concept in this concept lattice. So let's sat, we'll have a very small form of context that has four geometric figures. A square, a rectangle, a right angle triangle and actually an equilateral triangle, so all sides are equal. And we also have four attributes, attribute a means that a figure has exactly four angles, attribute b means that it has exactly three angles. C means that it has a right angle and d means that all sides in this figure, all the same, are equal. So for example, if we look at this figure, a square, we can see that it has exactly four angles. So it has attribute a, it also has equal sides. All sides are equal because it has attribute d and it has attribute c, so it has a right angle. Now let's look at two formal concepts, let's look at say this one And this one. So the first formal concept is In its extent, we have these three figures. So we have a square, A rectangle And a right angle triangle. And it's intent has only one attribute, attribute C. This is this concept. Now let's look at this one. This has, it's extent contains the same figures except for the right angle triangle so we have only square and the rectangle and its intent contains an additional attribute a. So in the intent we have ca. Now let's compute the supremum of these two concepts. Their supremum is just the concept that is more general. The first concept is the concept of all figures that have a right angle. And the second concept, is the concept of all figures with right angle and have exactly four angles. So this concept is less general than this one, and so this is the first one. If we want to compare the infimum. Then we can see that the infimum the one of the two which is the least general. So this one is less general than the first one, and so these are the infimum. A better concept. If the two concepts are comparable than the, The infimum is among them and the supremum is also among them. If the two concepts are incomparable, for instance let's look at this. Formal concept, And again at this formal concept. This one. So in the extent of the first one, we have a right angle triangle. And it's intent is b,c. So the properties are b c. And here we still have the same thing, a square, a rectangle and the intent is ca. Now if we compute the Supremum of this. The concepts, how can we see where the supremum is located in the diagram? We'll simply go along the upward arcs from out of this concept and we find the first concept where these two meet. It's going to be this one. So their supremum is this concept. The concept of right-angled figures. And if we want to compute the infimum Then we should go along the downward arcs before they meet. And they meet only in this node, in the bottom node. So this bottom node, Is the concept with the empty extent. And with the intended that contains all the four attributes. And this is not suprising, because here we'll have to find the cost of that called of the intersection of objects called by This concept and this concept. But this concept covers only right angle triangles and this concept covers only squares and rectangles. And there's no common object covered by both concepts. So we got this concept of nothing, the concept with an empty exempt. On the other hand here, we start from these two concept. And we. Their supremum includes all the objects covered by either this one or this one. So it includes square, rectangle, the right-angle triangle and the property, the attribute that they have in common. There is only one such attribute c. [MUSIC]