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However, from the Prime conditions it is not known Ideal Theorem. 1. The class X of inJeetive b o o l e a n the algebras has the properties. (I) 2 ~ X (2) X is closed (3) X is closed (4) X is closed Proof. 1 blish. (see Prin- result. 1 of (6) Lemma (4) is not very useful Applications algebra are to the Ordering Measure E x t e n s i o n of (I) can be found of (7) in  and 6. of these (I) follows of Semadeni under powers under restrictions. 6 ). under retracts (I), and (2) is an easy argument However, (3) and (4) are harder It is easy to see that u is a 2-submorphism to esta- iff u is an injec- 26 tion.
We now prove: 2 2T . By the remarks preceding the theorem, ~ is the Stone space of the Boolean algebra /~ of its clopen subsets with the usual set-theoretlc operations as Boolean operations. Moreover, is isomorphic to the power-set algebra First we show that ~ ~ ~ ~ (as sets) and ~(T). is a Boolean subalgebra of ~ ; the veri- fication of this fact is a matter of routine except, possibly, for the complement operation; (*) thus we prove: -~ x = -~ x Let #(vo,v~) be the formula: for every x c ~ . Vw[wEvo <'~ ~(wEv~) ].
07' to ~ " by adding a predicate Clb whose extension is ~ ; ~" = <~',~>. The following hold in ~ " : (i) Vv[Clb(v) (ii) Vvo[Clb(Vo) --~av~(©(v,) ~ Vz(zEv,<~- ZCVo))] -~ d~(v)] (every member of Clb is open); (the comp- lement of every member of Clb is open); (iii) Vv[Clb(v) ----~,(v)]; (iv) VVoVi[¢(Vo) A W(vl) A VoEV , ----*3vr[Clb(v~) a VoEV ~ A V z ( z E v 2 --~zEv,)]] (v) Vvov~[Pt(vo) (Clb iS a base); ^ Pt(v i) A ¢(Vo) ~ ~(v~) --* 3 W o W 1 [ ~ ( w o ) ~(w~) ^ ~(Wo) ^ ,(w~) ^ VoEWo ^ v~Ew~ ^ ~ z ( Z E W o (~ ^ ^ zEw~)]] is a Hausdorff space).