PS06 - MATH 681 Problem Set#6 This problem set is due at...

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MATH 681 Problem Set #6 This problem set is due at the beginning of class on December 3 . 1. (25 points) Given finite posets ( S, S ) and ( T, T ), let us consider the set S × T subjected to the relation which is such that ( a, b ) ( c, d ) if and only if a S c and b S d . (a) (5 points) Show that is a partial ordering on ( S × T ). (b) (5 points) Show that ( a, b ) is a maximal (or minimal) element of S × T if and only if a and b are maximal (or minimal) elements of S and T respectively. (c) (5 points) Show that if C is a chain in S × T , then the set C S of first coordinates of elements of C is a chain in S . (similarly, it is true that the set C T of second coordinates of elements of C is a chain in T ). (d) (5 points) Demonstrate that if A is an antichain in S × T , then the sets A S and A T defined as in the previous question need not be antichains. (e) (5 points + 5 points bonus) If S has height h S and width w S , and T has height h T and width w T , what upper and lower bounds can you put on the height and width of S × T ? Can you show that your upper and lower bounds are achieved
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