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B is said to preserve order if a ::; a' in A implies f(a) ::; f(a') in B. Prove that an order-preserving map fof a complete lattice A into itself has at least one fixed element (that is, an a e A such that f(a) = a). 3. Exhibit a well ordering of the set Q of rational numbers. 4. Let S be a set. ,t. 0, A C S. Show that the Axiom of Choice is equivalent to the statement that every set S has a choice function.

2(i) is true under these hypotheses. G =;t. 525 since e € G. 2(i). Thus a-I is a two-sided inverse of a. Since ae = a(a-Ia) = (aa-I)a = ea = a for every a € G, e is a two-sided identity. 1. 4. Let G he a sell1igroup. Then G is a group if and only if for all a, b € G the equations ax = band ya = b have solutions in G. PROOF. 3. • EXAMPLES. The integers Z, the rational numbers Q, and the real numbers R are each infinite abelian groups under ordinary addition. Each is a monoid under ordinary multiplication, but not a group (0 has no inverse).

Therefore H < (X) < H . • EXAMPLES. 8 (additive notation), ml = m for all m e Z. Of course the "powers" of the generating element need not all be distinct as they are in Z. The trivial subgroup (e) of any group is cyclic; the multiplicative subgroup (i) in C is cyclic of order 4 and for each m the additive group Zm is cyclic of order m with generator 1 e Zm. In Section 3 we shall prove that every cyclic subgroup is isomorphic either to Z or Zm for some m. Also, see Exercise 12. If {Hi lie I} is a family of subgroups of a group G, then of G in general.

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