 Hello and welcome to the session on the topic LATIS is under the course Discrete Mathematical Structures at Information Technology Engineering second year semester one. At the end of this session students will be able to classify algebraic systems as LATIS is. The flow of the session is LATIS is its definition we are going to learn new terms sub LATIS with its definition and examples and a direct product with its definition and examples. Now as we have already learnt LATIS as a partially ordered set pause for a while and try to answer this question. Can we consider a LATIS as an algebraic system so based on the answer we have one more definition coming along for a LATIS earlier we have defined LATIS to be an partially ordered set. Now as we have already learnt what is an algebraic system as well it is a system consisting of a partially ordered set along with two binary relations rather two binary operations. We can define a LATIS also to be an algebraic system and that will be a more useful definition when it is applicable to theory of computer science. So here is a definition for LATIS as an algebraic system a LATIS is an algebraic system L, star, plus with two binary operations star and plus on L which are both first commutative, second associative and third satisfy the absorption laws. Now again discussing about the definition first of all we observe that we have called this LATIS as an algebraic system so that is how the notation changes to earlier we only describe the LATIS to be a partially ordered set as L, some relation such as less than or equal to. Now here we have replaced the partial order relation by two binary operations namely star and plus and we say that these are the operations which hold the following properties and what are those number one commutative, number two associative and that too satisfying one of the important properties of LATIS is that we have learnt called as absorption laws ok. So this is how we describe a LATIS to be an algebraic system so based on this since we have described a LATIS to be an algebraic system we are now going to define a subsystem out of this which will be of more use for us and that will be termed as a sub LATIS. So here comes the next definition for a sub LATIS. Let L, star, plus be a LATIS and let S be a subset of L. The algebra or the algebraic system S, star, plus is called a sub LATIS of L, star, plus if and only if S is closed under the operations star and plus. So this is how we get a sub LATIS out of a LATIS so always the sub LATIS will be derived from some LATIS so here we say L, star, plus is a LATIS and then we define S being a subset of L a new algebra or an algebraic system S, star, plus. Note that the two binary operations remain same in the new algebraic system as well such as star and plus and then we say S, star, plus will be called as a sub LATIS if and only if there are two conditions to be satisfied that is both the operations star must be closed sorry the sets must be closed under both the operations star as well as plus. So now as earlier we have described we can replace the two operations namely star and plus by any two pair of operations such as less than or equal to and greater than or equal to or union and interaction intersection and dot and plus and so on and so forth. So these are only the notations that we use to denote some binary operation. So we derive a subset and then we say if both the new algebraic system is closed under two operations it is termed as a sub LATIS. Now the question arrives what do you mean by closed under these two operations? So we have learnt what do you mean by closed under some operation it is simply if you perform the operation of star as well as plus on any two elements of the given set the result that you get also belongs to the same set that is what we mean by closed under the operations star and plus that is if you take example of two elements A and B from the subset S and there is one more element C in the same set and we observe that if you perform A star B and that results into C and there is one more element D in the same set S and if you perform A plus B and that is equal to D which is also a part of S then we say since the result is also contained by the same subset S both the operations star and plus are closed under the two operations. That is simply whatever result you obtain after performing the operations you must find the element also present in the same set itself so that is how we define sub LATIS. So let us see an example of the same let L comma less than or equal to be a LATIS so here we have used earlier notation as a partial order set L comma less than or equal to be a LATIS in which L is equal to some set and S1 S2 S3 be the subsets of L given by S1 equal to A1 A2 A4 A6 S2 equal to A3 A5 A7 A8 and S3 is equal to A1 A2 A4 A8. Now we find S1 comma less than or equal to S2 comma less than or equal to sub LATIS of L comma less than or equal to here we must note that L contains certain elements A1 to A8 that is missing we write the elements of L as A1 A2 A3 A4 etc up to A8 and then we define these new subsets S1 S2 S3 so now S1 comma less than or equal to and S2 comma less than or equal to are sub LATIS of the given LATIS because both the operations star as well as sorry if you perform star such as A2 star A4 which results into A6 is also present in L but if you perform the same operation it is not an element of S3 so S3 comma less than or equal to is not a sub LATIS fine so but if you consider S3 comma less than or equal to as a separate algebraic system then you find that it satisfies all the properties of a LATIS so we say S3 comma less than or equal to is a LATIS. Next we have one more definition coming along which is again based on LATIS let L comma star comma plus and S comma disjunction comma conjunction as we have already said we can use any pair of symbols to denote certain operations so here we are using the disjunction and the conjunction B2 LATIS the algebraic system L cross S comma dot comma tilt these are again any two pair of symbols that we can use to denote in which the binary operations tilt and dot in L cross S are such that for any ordered pair A1 B1 and A2 B2 in L cross S we find A1 B1 dot A2 B2 is equal to A1 star A2 comma B1 conjunction B2 similarly if you perform A1 B1 tilt A2 B2 then you get A1 plus A2 B1 disjunction B2 this is called the direct product of LATIS is L comma star comma plus and S comma conjunction comma disjunction thank you