 Hello and welcome to the session on the course Discrete Mathematical Structures, the topic semi-groups and monoids under second year of information technology engineering semester 1. At the end of this session, students will be able to demonstrate semi-groups and monoids and its properties, these are the contents, semi-groups and monoids, definitions and examples, homomorphism of semi-groups and monoids. In the last video, we have learnt about algebraic structures, that is algebraic systems and its various properties, this is an extension of algebraic system where we learn semi-groups and monoids. So, let us understand what is a semi-group, here is the definition, let S be a non-empty set and dot be a binary operation on S, the algebraic system S comma dot is called a semi-group if the operation dot is associative. In other words, S comma dot is a semi-group if for any x, y, z which belongs to S, x dot y dot z is equal to x dot y dot z which is nothing but the property of associativity which we have also seen when we learnt about algebraic systems. So, now, here we can differentiate between an algebraic system being a semi-group, it is an algebraic system which possesses the property of associativity, a semi-group may or may not have an identity element with respect to the operation dot. We have seen examples of algebraic systems which also have an identity element for example the set of integers, set of natural numbers and so on. Next, monoid, here is the definition, a semi-group M comma dot with an identity element with respect to the operation dot is called a monoid. Now, what is a monoid? It is simply an extension of a semi-group and which holds the property of having an identity element with the respect to the operation dot. So, we can say every monoid is a semi-group which has an identity element. In other words, an algebraic system M comma dot is called a monoid if for any x, y, z which belongs to M, x dot y dot z is equal to x dot y dot z which is again the property of associativity which we have seen earlier in the semi-group definition and there exists an element e belongs to M such that for any x belongs to M we have e dot x which is the same as x dot e and which is nothing but equal to x. So, x is any element and e is an identity element for the monoid. A monoid has a unique or a distinguished element called the identity of the monoid. So, let us proceed with examples now, here are some of the examples which will make you understand what is a semi-group and a monoid in a better way. So, starting with the first example, let x be a non-empty set and x raise to x which we also called as x to the power of x be the set of all mappings from x to x. Let dot denote the operation of composition of these mappings that is for f comma g belongs to x raise to x f dot g given by f dot g of x is equal to f of g of x. Note the symbols that we have used for all x belongs to x in x raise to x. The algebra x raise to x comma dot is a monoid because the operation of composition is associative the identity mapping f of x equal to x for all x belongs to x is the identity of the operation and just now we have seen the definition for a monoid. A monoid is a semi-group which has an identity element and also a semi-group is an algebraic system where the operation with respect to the semi-group is associative in nature. So continuing with this manner, this example explains you what is a semi-group and what is a monoid. Let us see another example now. Example 2, let b of x denote the set of all relations from x to x and dot mean the composition of relations on b of x. Now the algebraic system b of x comma dot is a monoid in which the identity relation is the identity of the monoid and we all know what is an identity relation. So in this example what is an identity for this particular monoid here it is the identity relation itself which acts as an identity for the monoid b of x comma dot and also continuing with the semi-group definition b of x comma dot is also a semi-group where the dot operation is associative. Now make an assignment where you will try to find out this question. Let n be the set of natural numbers then define algebraic systems n comma plus and n comma into where plus is the operation of addition and into is the operation of multiplication. What are these semi-groups or monoids? Here is the answer, both the systems n comma plus and n comma into are monoids with the identities 0 and 1 respectively. I hope you can understand the identities being 0 for addition and 1 for multiplication as any element from set of natural numbers if you perform addition with 0 it gives you the same number itself. Similarly, if you perform multiplication of any element of set of natural numbers with 1 it gives you the same number itself. So it is clear that for addition the identity element is 0 which also belongs to the set of natural numbers and for the operation of multiplication the identity element is 1 which also belongs to the set of natural numbers and obviously these are also semi-groups as both the operation of addition as well as the operation of multiplication are both associative in nature. Lastly, consider this example if capital E denotes the set of positive even numbers then what can be said about these algebraic systems e comma plus and e comma into algebraic systems namely e comma plus and e comma into are only semi-groups because they only satisfy the property of associativity. Since it contains only the positive even integers or positive even numbers while finding an identity for addition it is difficult to find in such similarly finding an identity for multiplication from set of positive even numbers it is not possible. Now as we have learnt various properties of an algebraic system same we are going to learn in terms of semi-groups as well as monoids starting with the first property being that of homomorphism definition let s comma star and t comma delta be any two semi-groups a mapping g from s to t such that for any two elements a comma b belongs to s g of a star b is equal to g of a delta g of b is called a semi-group homomorphism just like algebraic systems a semi-group homomorphism is called a semi-group monomorphism if it is the mapping is one to one it is called as a semi-group epimorphism if the mapping g is an onto and it is called as a semi-group isomorphism if the mapping g is one to one onto so similar to the algebraic system the same applicable to semi-groups. Now two semi-groups s comma star and t comma delta are said to be isomorphic if there exists a semi-group isomorphic mapping from s to t this is also similar to the algebraic system definition that we have learnt isomorphic images of each other so same s comma star and t comma delta are isomorphic mappings from s to t but in case of a semi-group next comes a definition for homomorphism of monoids let m comma star comma e of m and t comma delta comma e of t be any two monoids where e of m denotes the identity for first algebraic system and e of t denotes the identity for the second a mapping g from m to t such that for any two elements a comma b belongs to m g of a star b is equal to g of a delta g of b and g of e of m is same as e of t what does that mean when you apply the definition of identity element we find that the mapping g of applied on the identity element of the first set m gives you the identity element for the second now this is called a monoid homomorphism so similar to the semi-group homomorphism we have a similar condition given by g of a star b equal to g of a delta g of b and we have an additional condition where it relates to the identity elements of the two sets g of e m must be same as e of t a homomorphism of a semi-group into itself is called a semi-group endomorphism why an isomorphism onto itself is called a semi-group automorphism these are also in line with the definitions of same in case of algebraic system these are the references in the next video we are going to learn more about semi-groups and monoids with respect to sub-semi-groups and sub-monoids thank you