 Hello and welcome to the session on the topic LATIS under the course Discrete Mathematical Structures at second year of Information Technology Engineering semester 1. At the end of this session students will be able to classify algebraic systems as LATIS. The flow of the session is as follows, we will start by deriving the definition of a LATIS followed by the demonstrating examples of LATISes which will throw more light on what is LATIS as an algebraic system. Before moving ahead, let us pause for a while and try to answer this question by recollecting whatever we have learnt in set theory before. The question is, what is a partially ordered set having both GLB and LUB called? So, with this let us try to define a LATIS, here is the definition a LATIS is a partially ordered set L, less than or equal to in which every pair of elements a, b which belongs to L has a greatest lower bound denoted by a star b also called as a meat or product and a least upper bound denoted by a plus b also called as join or sum, star and plus are binary operations on L. So, we start with what is a LATIS? LATIS is basically a poset, so here we say a totally ordered set is a LATIS but not all partially ordered sets are LATISes, this is just an important observation where we have learnt more about partially ordered sets being a totally ordered set, with this we will see certain examples which will explain what is a LATIS actually and then we will learn more about the properties which it holds depending upon the binary operations. So, here comes the first example, the examples are also taken from what we have learnt earlier like set theory, the relations, the partial order relations and so on, so the first example is of the power set, I hope you remember what is a power set, it is a set containing all possible subsets of a given set. So, let S be any set and row of S be its power set, the notation that we use, the Greek letter that we use to denote the power set, row of S which contains all possible subsets of given set and also note that if there are n number of elements, we simply get 2 raised to n number of possible subsets including phi the empty set and the given set itself. Example says the partially ordered set, row of S, lesson equal to is a LATIS in which the meet and join are the same as the operations, intersection and union respectively. So, when S has a single element, single element as in the number of elements present in the given set S is equal to 1, so in this case we have n equal to 1, so in that case we will have the number of subsets possible will be 2 raised to 1 2 which will be phi the empty set and the given set itself containing the single element. So, we say the corresponding LATIS is a chain containing 2 elements which are those 2 elements, number 1 the phi the empty set and second the given set itself containing the single element. So, that is how if you reconnect this to the statement saying that totally ordered sets are always LATIS because they represent a chain. So, here again in particular when S has a single element the corresponding LATIS is a chain containing 2 elements. Here is the second example, now we consider this from the number system let us say we say let I plus be the set of all positive integers and let D denote the relation of division in I plus such that for any a comma b in I plus we write the ordered pair A D B this is the notation that we use A is related to B by the relation D that is division. If A divides B largest number divides both A and B simultaneously and it is the greatest number. So, this is a list of properties these are some of the properties. So, starting with L 1 A star A is equal to A A plus A at the same time for the other operation is also equal to A and this property is termed as the idempotent property. I hope all of you remember this. Second L 2 A star B is equal to B star A and A plus B is equal to B plus A this is termed as a commutative property. L 3 A star B star C is equal to A star B star C and same is applicable for the other operation and this we term as the associative property. We have learned these properties so many times earlier when we learnt about the statement algebra as well as the set algebra fine. And lastly L 4 denotes A star in bracket A plus B where the two operations namely meet and join both are involved. So, you can observe A star A plus B is equal to A B is being absorbed similarly L 4 dash A plus A star B is equal to A B is again getting absorbed. So, we simply call this property as an absorption property. So, these are the four main properties which are lattice holds idempotent commutative associative and absorption. And with their usual meanings as we have already learnt in case of the earlier theories. And we say once you define L 1 to L 4 the properties or identities L 1 dash to L 4 dash automatically follow from a principle of duality is it not. We have learnt this principle earlier where in terms of sets we say a dual of a formula is obtained simply by replacing conjunction by disjunction and vice versa. So, here the two operations under consideration are named as meet and join. So, we simply replace a meet by join and vice versa. So, as to get the other identity right from L 1 dash to L 4 dash fine. So, these are some of the properties of a lattice. Now, here is an assignment for all of you to solve. We have seen three examples. Now, here is one more to solve for you all. Let S be a non-empty set and P of S be the set of all partitions of S. So, again a concept which we have learnt in terms of the relations partitions and covering if you remember or other sets what is a partition of a set and also covering of a set. So, partition of S and then let S is given by a simple set containing three elements ABC define a lattice for the given set. So, what you are supposed to define here is the lattice as a partially ordered set with the set and the relations such as less than or equal to. Then you define the two operations namely the meet and the join operation and then you list out certain properties of these two. Thank you.