(1) Prove that all monomial idenls alrendy have Gröbner bnses....

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(1) Prove that all monomial idenls alrendy have Gröbner bnses. (2) A fully supported binomial in R = k|re is a polynomial of the form Xa- XB where o +8 has no zero terms and a and 8 share no nonzero terms. For example, 1433 - is a fully supported in k(211, rs). Let I C Art ,Inl be an ideal generated by fully supported binomials. Prove that if G is n Gröbner basis of I, then G contains all the binomial generntors of 1, plus possibly more generators. (3) Prove that n set A C N² is an antichain if and only if the slope between any two elements of A is negative. (4) Which of the following sets is an antichain? If it is not an antichain, what is the largest antichain in the set? Is that largest antichain unique? (n) A = 2,3), (1,8,7), (2,1, 1), (7, 4), (1,8,2), (3,7,1)} (b) A = {(5,4,3, 2), (6,1 4, 1), (8, 7, 2), (6, 3,3,2), (1, 1,1,9)} (c) A = {(3, 2,5), (4, 1, (6), (5,8,2), (6, 3, 1), (7, 4,2)} (5) Draw the Buchberger graph for the following ideals. (n) I = (b) I = (c) I (6) Write the resolution for ench idenl in number 5. 1

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