1. Let D be the set of finite and infinite sequences over Sigma := {a,b} with c= the prefix ordering. a. Verify that this is a domain b. Which of the following functions f:D->D is monotone / continuous? (i) f(s) = s with all a's removed (ii) f(s) = abba if s is finite; f(s) = s if s is infinite (iii) f(s) = abbas (iv) f(s) = a if s contains finitely many b's; f(s) = b if s contains infinitely many b's c. Compute the least fixed point of the f in (b) that are continuous. 2. Let D be a domain with element d and let f be a continuous function from D to D. Suppose d <= f(d). Prove that the lub of the chain f^i(d) (i in IN) is a fixed point of f. 3. For the "binary sum of domains" (also called the "disjoint union" of domains), there are two choices: the "smashed sum" D +s E and the "coalesced sum" D +c E. For the smashed sum, the set D +s E is defined as {bot} U {(0,d) | d in D, d <> bot_D } U {(1,e) | e in E, e <> bot_E} For the coalesced sum, the set D +c E is defined as {bot} U {(0,d) | d in D} U {(1,e) | e in E} So, the coalesced sum introduces a new bot element, whereas the smahes sum "smahes them together". For both definitions of sum: (a) define the ordering <= on D+E and check that it is a po (b) define a bottom element of D+E and check that it is least (c) for (d_i) a chain in D+E define the lub of (d_i) and prove that it is the least upperbound (d) define injections inl : D -> D+E and inr : E -> D+E that are continuous. (You don't have to prove that they are continuous.) (e) for F a domain and f : D -> F, g : E ->F define a continuous function [f,g]: D+E -> F such that [f,g](inl x) = f(x) and [f,g](inr x) = f(x). (You don't have to prove that it is continuous.) Note: For one of the definitions of D+E this can only be done if we pose and additional requirement on f and g. Which?