[MUSIC] There are some things we can do to simplify formal context without changing the structure of the concept lattices. One such thing is called clarification. Well, the idea is there if you have two identical objects in your formal context, you can remove one of them, and you'll get the same concept lattice. Well, the concept lattice will not be exactly the same. But the structure will look the same. You'll get the same number of formal concepts. They will be connected to each other in the same way. But their extents might be different because you removed one of the objects. Still the structure is the same. And so we say that a context, a (G, M, I) is clarified, If you don't have identical objects and you don't have identical attributes. So g prime is different from h prime whenever g and h are two different objects from g. And also m prime is different from n prime whenever, m and n are two different attributes from m. So that's an easy modification. You identify objects with identical intents and you keep only one of them. You identify attributes with identical extents and you keep only one of them. But there is more you can do. Remember, we were talking about irreducible elements. So in context, we also have reducible and irreducible elements. An object, g, but now, let's say that we work with a clarified context. So we're looking at an object g in the clarified context. A context that doesn't have objects with identical intents and it does have attributes with identical extents. So we're already done the first operation, clarification. And I would say that an object in such a clarified context is reducible, If there are other objects. There is a set, subset of objects S, such that and this set doesn't include g. Such that g prime is the same as S prime. So if you compute the intersection of all object intents from S, then you'll get the intent of g. In this case, g is reducible. In a sense, you don't need it. Well, what's not needed? You don't need this object to obtain an element in the lattice that corresponds to it. Because you can obtain it using other elements, those from S. And a similar definition exists for attributes. So we say that an attribute, M, In a clarified context, Is reducible, If we can find a subset of attributes, Which doesn't include m, Such that m prime equals to S prime. So again, the [COUGH] extent of the attribute m is simply the intersection of attribute extends from S. So if an object or an attribute is reducible, we, in a sense, don't need them. We don't need them to obtain the same structure of the concept lattice. We might need them for some other calculations. For example, if want to do some statistics, then every object is maybe valuable. We can't really remove an object and forget about it completely. But if all we are interested in is the structure of the concept lattice, then we can do without reducible elements and keep only those that are irreducible. And this process, the process of removing reducible elements, is called context reduction. So let (G, M, I) be a clarified context. And let's identify objects that are irreducible. Denote them by Girr. So Girr is the set of all irreducible objects. And Mirr be the set of its irreducible attributes. Then the context, Girr, Mirr. So the context with the irreducible objects from our original context. And with only irreducible attributes from our original context and the relation is simply restricted to the Cartesian product of Girr and Mirr. So we take I intersection, Girr, Mirr. Now this context is called the reduced context corresponding to (G, M, I). Again, we start with (G,M,I). We keep only irreducible objects, we can keep only irreducible attributes, and so the relation simplifies in a natural way. What we get is a reduced version of the original context and the claim is that the concept lattice of this context is isomorphic to the concept lattice of (G,M,I), it has the same structure. This is a useful property because the context might have lots of objects. And it may be a little bit difficult to do any serious computation when you have lots of options. It may be even difficult to build a concept lattice. So one thing you can do to simply your context is to clarify it. Remove identical objects and attributes. But then you can make it small even further if you apply reduction, if you remove reducible objects and attributes. There are more or less efficient procedures to do this. And once you have the reduced version of your formal context, you may do other computationally intensive operations. Such as you can construct the concept lattice of your formal context. And this will take less time than if you do it directly on (G, M, I). So one other thing. Now, let's suppose that l, Is a finite lattice. Let's say that j(L), is the set of its supremum irreducible element. And similarly, M(l) is the set of infimum irreducible elements. Now let's look at the formal context J(L), M(L), and less than or equal to. So this is the relation that comes from all lattice L and we use it as the incidence relation in our formal context. This is called, The standard, Context of all lattice L. And one thing we can be sure of is that it is a reduced context. We started, we took only irreducible elements as objects and as attributes. So this is a reduced context. And the concept lattice of this context is isomorphic to L. In fact, in the previous video we used precisely this method when we wanted to build a formal context corresponding to the lattice from which we started. We took supremum irreducible elements, took them as objects. Then infimum irreducible elements, took them as attributes. And we obtained a context which is, as it turns out, the smallest possible context for this particular lattice. And it's called the standard context of the lattice. In the next video, let's look at some examples. [MUSIC]