 Hello and welcome to the session on the topic lattices under the course discrete mathematical structures at second year of information technology engineering semester one. At the end of this session students will be able to classify lattices into different types. We have learnt lattice as a partially ordered set first and then we have also learnt a lattice as an algebraic system. Then we have learnt the different types based on lattices such as sub lattices and so on. Now in this the flow of the session will be understanding the special types of lattices such as complete, complemented and distributive followed by their definitions along with some examples. Now pause the video for a while and try to answer this question. How can we classify lattices into different types? As a need of a theory in computer science says we must divide or classify a lattice into different types based on some criteria. So can you think of this criteria based on which we can classify or categorize the lattices? So it is the answer. We start with a complete lattice and if you observe from the definition we will come to know that we have categorized the lattices depending upon whether they contain the important elements which we have also used while defining a lattice as an algebraic system termed as LUB that is the least upper bound and GLB that is the greatest lower bound. So here is a definition for complete lattice. A lattice is called complete if each of its non-empty subsets has a LUB and a GLB. So we have defined a lattice to be complete lattice based on if you assume any non-empty subset derived from a lattice it must contain both LUB and GLB in that case the lattice will be termed as a complete lattice. And this is because while defining the lattice also we have started with saying that a lattice is the one which contains both LUB and GLB and then if you are taking a subset out of this lattice to become a lattice once again or calling the lattice to be a complete lattice such that every subset of it will contain both LUB and GLB then the lattice will be termed as a complete lattice ok. So the property which it holds for itself that is having an LUB and GLB is true for every possible subset of it having both LUB and GLB. Since it carries forward the property we can term the lattice to be a complete lattice fine. Here is the second type complemented lattice so we say a lattice L with the binary operations and the elements called as 0 and 1 of a lattice is said to be a complemented lattice if for every element of L it has at least one complement and we know that what is the complement such that if you perform the operation of either of the binary operations given in the lattice such as in this case it is a star or a plus it results into the 0 or 1 of the element. For example if I perform for any element such as a and b a star b is equal to comes out to be 0 then I say b is called as a complement of a. Similarly if I perform a plus b and that results into 1 I say b is termed as a complement of a. So the second classification is based on having a complement for every element from the given lattice. So here is the definition for a complemented lattice L is said to be complemented if for every element present in L it has at least one complement there is no restriction on the number of complements present but it must have at least one which is the minimum requirement to be called the lattice to be a complemented lattice. So we started with the categorization based on having a LUB or GLB so that is termed as a complete lattice and here is based on whether it contains a complement so we call the lattice to be a complemented lattice. So on the same line we can go for one more category of lattice and what is that yes see when we defined a lattice as an algebraic system we said that it contains a property of commutative and associative but we have not talked about the third important property that is a distributive property. So now if you are able to satisfy for the given lattice the property of distributivity also then we get a different type of lattice because along with being commutative and associative now it holds the property of distributivity as well. So here comes the third type of a lattice based on whether it satisfies the distributive property. So a lattice L with two binary operations star and plus is called a distributive lattice if and only if for any A, B, C in L you consider three elements from L and if you perform the operation such as A star B plus C and that comes out to be A star B plus A star C or you perform A plus B star C and that results into A plus B star A plus C. So if you notice what we have applied is the distributive law for the two binary operations star and plus so that is why the name given to the lattice is simply the distributive lattice. Why we have come up with this third type is because there are certain algebraic systems where the property of commutativity as well as associativity holds and at the same time if you are able to satisfy for the given two binary operations the property of distributivity also then it may result into something else and that we have classified as a distributive lattice. So from the name itself we come to know that it is a lattice which along with the earlier two as well as if you recollect it also satisfies the absorption law. So while defining a lattice initially we have said that it is commutative associative and also satisfies absorption law. So along with these three now if you add distributive law or property it results into a distributive lattice. So once again revising the three types of a lattice first one which is dependent upon having either GLB or both LUB and GLB we call it as a complete lattice. Second if every element has at least one complement present in the same lattice it is termed as a complemented lattice and the third category if along with the earlier properties of commutativity, associativity and absorption if it holds distributive property as well the lattice is termed to be a distributive lattice. Thank you.