 Hello and welcome to the session on sub-semi-groups and sub-monoids under the course Discrete Mathematical Structures at 2nd year of Information Technology Engineering Semester 1. At the end of this session, students will be able to demonstrate sub-semi-groups and sub-monoids and its properties. These are the contents we are going to cover, sub-semi-groups and sub-monoids with the help of their definitions and number of examples. To start with, in the last session, we have learnt about semi-groups and monoids which are types of algebraic structures. Now in this session, we are going to understand what are sub-semi-groups and sub-monoids. So to start with, here is a definition for sub-semi-group. Let S, star be a semi-group and T is any set which is a subset of S. If the set T is closed under the operation star, then T, star is said to be a sub-semi-group of S, star. Now let us discuss the definition in detail. We have learnt the definition of a semi-group. So here, S, star is a semi-group where S may be any set and star is any operation. Now to define a sub-semi-group, we have a new set defined as T and which happens to be a subset of the given set S. We further define T is a set which is closed under the operation star. Now by saying that it is closed under the operation star, we mean that whenever we perform the operation star on any two operands from the set T, the result that it generates, it also belongs to the same set T. So that is what we mean by closed under the operation star. Now let us understand with the help of some examples. Here is example number one. For the semi-group N, into that is N, multiplication where N is a set of natural numbers, let T be the set of multiples of a positive integer M. Then the algebraic structure T, into is a sub-semi-group of N, into. Now here to apply the definition we can assume all the operations that we perform on the elements those are multiples of the positive integer M. So it is obvious that the result that it generates with the multiplication operation of any two positive integers is going to be contained by the same set T and that is what we mean by closed under the operation of multiplication. So I hope the example makes you clear how do you find a sub-semi-group of a given semi-group. Here is another example, example number two. For the semi-group N, plus where plus is the operation of addition, the set E of all the even non-negative integers is a sub-semi-group E, plus of N, plus. Once again let us understand how E, plus becomes a sub-semi-group. Now we know that since E is a set of all even non-negative integers and N which is a larger set and it is set of natural numbers, obviously whenever you perform addition of any two even non-negative integers the result is going to be contained by the same set E and obviously all these are present in set of natural numbers N hence E is a subset of N. So it is exactly applicable to the definition of a sub-semi-group hence proved. Now pause the video for a while and try to answer this question. Now since we have learnt about semi-groups and based on that we have learnt about sub-semi-groups and we have also learnt about a monoid, now can we also define a sub-monoid? Think for a while and try to devise the definition for a sub-monoid on your own. Here is the answer, the definition for a sub-monoid. Let M, star, E be a monoid and we know that M is any set, star is an operation and E is an identity element with respect to operation star. So if you recollect we have defined a monoid based on the definition of a semi-group where if a semi-group has an identity element then that semi-group is called as a monoid. So back to the definition of a sub-monoid let M, star, E be a monoid and T is any set once again which happens to be a subset of the given set M. Now if the set T is closed under the operation star and E belongs to T then T, star, E is said to be a sub-monoid of M, star, E. Now if you notice the definition once again considers T is a set which is closed under the operation star it is same as the sub-monoid and sub-semi-group and additionally we have an identity element E which also happens to be present in the set T and then we define T, star, E which is an algebraic system and it is said to be sub-monoid of M, star, E. So how it differs from the earlier definition of a sub-semi-group is here is an additional condition being that of identity element E which must be also present in the set T. Now moving ahead with some more definitions about sub-semi-groups and sub-monoids here is a first definition. Let S, star and T, delta be two semi-groups. We define the direct product of S, star and T, delta is the algebraic system XT, dot in which the operation dot on S cross T is defined by S1, T1, S2, T2 is equal to S1 star S2, T1, delta, T2. Now if you notice in between these two pairs of S1, T1 and S2, T2 there is no operation that it happens to be the operation of direct product and it is defined for any S1, T1 S2, T2 which are part of or which belong to S cross T. Now with the usual meaning of cross it considers the cartation product of the two sets S and T and we know that whenever we perform a cartation product it assumes all possible pairs that comes from the two participating sets namely S and T. So once again going back to the definition we start with two semi-groups defined as earlier S, star and T, delta where star and delta can be any two different operations and we define the direct product which comes out of these two semi-groups as S cross T and it holds the condition of being whenever you have a pair S1, T1 performed as direct product with S2, T2 it is equal to the first operation star performed on the first two elements S1, S2 and the second operation delta performed on the other pair T1 and T2. The direct product of any two semi-groups is also a semi-group that is whenever we perform direct product of any two given semi-groups it comes out to be a system which is also a semi-group. Now if S, star and T, delta are both commutative semi-groups then their direct product is also commutative. Now being commutative we know that the prop so the result is not going to be changed it is same as the earlier one that is S1 star S2, T1, delta, T2. So obviously whenever these two participating entities are commutative then their direct product has to be also commutative in nature. Moving ahead, here are some more definitions. If S, star and T, delta are monoids with ES and ET as their identity elements respectively then their direct product S cross T comma dot is also a monoid with ES comma ET as identity element because ES comma ET dot S comma T is equal to ES star S comma ET delta T which is equal to S comma T. Now which is quite obvious I hope you understand because ES and ET are the identity element and since these are identity element when you perform the operation with these identity elements of any elements present it gives you the same result. So that is why if you concentrate on the right hand side ES multiplied by S is equal to S ET multiplied by or rather the operation delta with T also gives you the same result as T and the second part says the pair S comma T dot ES comma ET is equal to S comma ES S star ES comma T delta ET which is again same as S comma T. So what we have done in the second part is we have also shown that it is commutative the earlier operation dot which happens to be commutative also generates the same result with the identity elements ES and ET. So with this we have covered what do you mean by a direct product of subsemi groups and also a direct product of sub monoids. These are the references. Thank you.