For which n E N does the set of all positive divisors of n form a chain, with a :::: b

The set of all nonzero integers is not partially ordered with a :::: b defined to mean a lb. Why? (Compare Example 63.4.)

Draw the diagram for the set of subsets of {x. y 1 with the relation <;.

Draw the diagram for the set of subsets of {w. x. Y, z 1 with the relation <;.

Draw the diagram for the set of positive divisors of 20, with a :::: b defined to mean a I b.

Assume that p and q are distinct primes, and let n = p2q. Draw the diagram for the set of positive divisors of n, with a :::: b defined to mean a I b. (Compare Figure 63.2 and Problem 63.4.)

For which n E N does the set of all positive divisors of n form a chain, with a :::: b defined to mean alb?

For N partially ordered as in Example 63.4, determine the l.u.b. and the g.l.b. of the subset {12, 30, 126}.

Consider the partially ordered set N with a :::: b defined to mean a I b (Example 63.4). (a) Which integers are covered by 67 (b) Which integers cover 6? (There are a lot.) (c) For mEN, which integers are covered by m? (d) For mEN, which integers cover m?

Consider the partially ordered set of all subgroups of the group of integers (Examples 63.3 and 63.10 with G = Z). (a) For which m, n E Z is (m) <; (n)? (b) For m, nEZ, determine k so that (k) is the greatest lower bound of (m) and (n). (c) For n, m E Z, determine k so that (k) is the least upper bound of (m) and (n).

Assume that A and B are subgroups of a group G. Prove that A U B (set union) is a subgroup iff either A <; B or B <; A. (This explains why (A, B) rather than A U B is used in Example 63.10.)

A mapping 8 : L 1 ...... L2 of one partially ordered set onto another is called order preserving if a :::: b implies 8(a) :::: 8(b )for all a, bEL 1 An invertible mapping 8 : L 1 ...... L2 is an isommphism if both 8 and 8- 1 are order preserving. (Finite partially ordered sets are isomorphic iff their diagrams are the same except possibly for labeling.) (a) Verify that isomorphism is an equivalence relation for partially ordered sets. (b) There are two isomorphism classes for partially ordered sets with two elements. Draw a diagram corresponding to each class. (Remember that two elements need not be "comparable.") (c) There are five isomorphism classes for partially ordered sets with three elements. Draw a diagram corresponding to each class. (d) There are 16 isomorphism classes for partially ordered sets with four elements. Draw a diagram corresponding to each class. (e) Formulate a necessary and sufficient condition on positive integers m and n for the partially ordered sets of positive divisors of m and n to be isomorphic, with a :::: b defined to mean a I b. (Suggestion: See Problem 63.5.) Give several examples to support your condition, but do not write a formal proof.

Prove that IP(S)I = 21sl (See Example 63.2. Assume lSI = n and x E S. The subsets of Scan be put in two classes, those that do contain x and those that do not. The induction hypothesis will tell you how many subsets there are in each class.)

Prove that the subgroups of every cyclic group of prime power order form a chain, as claimed in Example 63.11.