By Carole E. Greenes

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**Example text**

For let is dense in w h i c h is impossible Hence Lemma 2 gives that T Tla and since is Souslin. | Theorem Assume ~. Then there is a Souslin tree with exactly wI automorph- isms. Proof: We show that the h o m o g e n e o u s has exactly for wI a,~ < w I , Souslin tree automorphisms. a # ~ . So T Of course, has at least The converse direction is an immediate the f o l l o w i n g Claim: If c T constructed c[a]~T wI consquence above f o[~]~T automorphisms. of CH and claim. is a u t o m o r p h i s m of T, then there is an a < mq - such that for all then x E T - and v < ~ if , m ~ ~ < ht(x) , c(x)~ = x v .

2. The details enm: ~ n - , ~ m is its direct the system 2 : ~ and a commutative fined by composition). "l~o Proof: set dom~B2) for example, of the proof clearly Let simply of but for and e . We shall (in the sense is obvious, Let and (by normality) b o E ~o Lemma ~2 ' use llz = x v y Given Lemma - is just lemma 85 of [Jel]. construction many 57 ~w on be its as a poset of the chain and - (~n] n < w ) identity, plete we call the embeddings en (~Bn)n~ here In the case where BA , will . the embeddings with , the direct Suppose algebra - , together :~n-~ ~ (enm)n

CJ' w-sequence (enm I n ~ m ~ is the identity algebra ~ M M successively BA enm If , then (where, This is a boolean (One way = BA(~) ) . in A natural so that each = Un~w~ n completion. to deal union. 2. The details enm: ~ n - , ~ m is its direct the system 2 : ~ and a commutative fined by composition). "l~o Proof: set dom~B2) for example, of the proof clearly Let simply of but for and e . We shall (in the sense is obvious, Let and (by normality) b o E ~o Lemma ~2 ' use llz = x v y Given Lemma - is just lemma 85 of [Jel].