 and this is joint work with Dr. Krithi Gowel and Professor Adrenal Sengupta. By an affine semi-group, we mean a finally generated additive sub monoid of ND for some positive integers D. The cardinality of minimal generating set of an affine semi-group is known as the embedding dimension of S and it is denoted by E S. So, let S be a semi-group minimally generated by A1 up to An which are vectors in ND. The semi-group ring S is a case of algebra of the polynomial ring generated by the monomials T to the power A i where T A i is defined like this. Set R equals to polynomial ring and define a map pi from R to K S given by pi x i equals to T to the power A i and set degree x i is A i. Then this is a multigraded ring and pi is degree preserving subjective K algebra homomorphism and let kernel of pi is I S, then I S is a homogenous ideal and generated by binomials and it is called the defining ideal of S. Now, consider the cone of S in the rationales which is defined as the rational linear combinations of generators of the semi-group and set this at H S which are the cone S minus S and intersect with ND. An element from H S is called pseudofovenous elements if F plus S belongs to S for all non-zero S from the semi-group and this set is denoted by PFS and we call the cardinality of this set the Betty type of S. Remark the pseudofovenous elements may not exist. For example, let S is the semi-group generated by 2 0 1 1 0 2 then this S is the subset of points in N2 whose sum of coordinates is even. It means the H S is the subset of points in N2 whose sum of coordinates is odd and if we add S in H S then we will again lend into H S. Therefore, PFS is empty but when H S is finite and non-empty then this set PFS is always non-empty. Now, consider the partial order on ND defined as x is less than equals to y if y minus x in S. With this partial order PFS is the maximals of H S. For example, let this semi-group S. So, here this big green points are generators and the green points and the points under the shaded region are the points of semi-group and red points are the set of H S and you see these the points this one and this one these are the maximum of H S and these are the pseudofovenous elements. Now, we define the maximal projective dimension semi-group. So, semi-group S has maximal projective dimension if its projective dimension is N minus 1 where N is the number of generators equivalently depth of K S is 1. In Garcia et al. in 2020 proved that S is MPD if and only if the set of pseudofovenous elements is non-empty. Also they proved that if S is a MPD semi-group then A in S is the S degree of N minus 2th minimal Cgc if and only if A belongs to this set and this cardinality of this set is equal to the last betty number of K S over R. For example, let S is generated by 2 11 3 0 5 9 7 4. Then by Macaulay 2 we have this resolution minimal free resolution and the graded betty numbers of 2nd Ciszi modular 81 93 and 94 82. Then by the previous theorem of Garcia we see the PFS is 81 93 minus sum of the generators and 94 82 minus sum of generators and this set. So, here see the H S is not finite. So, it is difficult to find PFS from an infinite set. So, we can find PFS with the help of minimal free resolution. Now, we define symmetric semi-groups. Let press be a tau mod of N D and F S is the maximum of H S with respect to this order if it exists is called the Frobenous element of S. So, Frobenous element may not exist, but if H S is finite then they always exist. Fix a tau model and suppose F S exist. If PFS is F S only then it is called symmetric semi-group and if F S and F S by 2 then it is called pseudo symmetric semi-group. So, when H S is non-empty finite then it is called a C semi-group. We have C denotes the cone of semi-group and for C semi-group and when semi-group is symmetric and pseudo symmetric we give a characterization of this semi-group. Let us see a C semi-group and F S denotes the Frobenous element with respect to an order then S is symmetric semi-group. If and only if for each G in cone S intersects an N D we have G in S if and only if F S minus G not in S and it is similar characterization for pseudo symmetric. Now, we define extended Wilsk conjecture. The Wilsk conjecture is defined for numerical semi-group by Wilsk in 1978 and it is in since C semi-groups are natural generalizations of numerical semi-group in the sense that complement of natural numbers of numerical semi-group is finite and here complement of C semi-groups in the cone is finite. So, every numerical semi-group is also a C semi-group. So, in the C semi-group we define the Frobenous number like this and F S is cardinality of H S plus cardinality of the set G in S such that G is less than F S with respect to a tau model. So, Garcia et al extended the Wilsk conjecture to the C semi-groups like this. The Frobenous number plus 1 is always less than equals to the product of embedding dimension and the product of and the cardinality of this set. So, for numerical semi-groups in natural numbers this is just that Frobenous number the largest which is not the semi-group and this is just the elements which are not gap, but less than equal to the Frobenous L. So, less than equals to C is the usual relation on ND. Usual relations means G is less than equals to F if the all the components are less than equals to. So, for the C semi-groups with full cone we proved that S is symmetric if and only if the cardinality of these two sets are equal H is pseudo symmetric if and only if cardinality of these two sets are equal. With the help of previous characterizations we prove that for sigma symmetric or sigma pseudo symmetric semi-groups the extended Wilsk conjecture holds. For example, let look at this example. Let S be the semi-group defined by generated by 3 0 5 0 0 0 1 1 3 and 2 3. Then here we can see 7 2 this is the maximum with respect to the degree lexicographic order and we see H s is these elements which are not pointed. So, cardinality of H s is 12 and the cardinality of this set G in S G is less than equals to C F s is also 12 this one and we see emerging dimension is 5 and N F s is the sum of cardinality of H s 12 and those elements which are less than equals to F s with respect to that order degree lexicographic and that is 41. Then the Frobenius number is 53 and product of E S and this is 60 then 53 is less than 60. Then it is also less than the product of this is also less than the product of this and this because this is the bigger set. So, less than equals to C order is also is a smaller one. Now, we define the gluing of semi-groups. So, let G S be the group generated by the semi-group and A be the minimal generating system of S and the partition the A into two parts A 1, U and A 2 this is a non-trivial partition. Let S I be the submodents of N D generated by AIs. Then S is S 1 plus S 2 we say S is the gluing of S 1 and S 2 by S if S is in the intersection of S 1 and S 2 and the intersection of groups generated by S 1 and S 2 equals to the S z means generated by S only. So, for the gluing of semi-groups we proved that S is MPD if and only if S 1, S 2 are MPD and this is the description of pseudo Frobenius elements of gluing. So, the pseudo Frobenius elements of gluing are look like this. So, the question arises this set is PFS is always finite this set PFS is always finite, but the question is whether this is bounded in terms of embedding dimension. So, the answer is no we prove by an example that where the embedding dimension is always 4, but the cardinality of this set is increasing. So, let A is greater than equals to 3 an odd natural number and let P be a positive integer define this semi-group SAP generated by 4 elements and define this z delta which is the cardinality of this set is dependent on A and P. Then we prove that SAP is an MPD semi-group and this z delta is contained in the pseudo Frobenius elements and by the technique of gluing we prove that for each E greater than equals to 4 there exists a class of MPD semi-groups of embedding dimension E and N2 where there is no upper bound on the body type in terms of embedding dimension E.