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Cardinal Invariants on Boolean Algebras by J. Donald Monk

By J. Donald Monk

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"This publication is an essential instrument for somebody operating in Boolean algebra, and can also be prompt for set-theoretic topologists." - Zentralblatt MATH

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20. 22. Let A ::; B, let f : B --+ Q be an epimorphism, and set P = f[A]. Then A :=:::m B implies that P :=:::m Q. If, moreover, kerf~ A, then A :=:::m B iff P :=:::m Q. 34 2. Special classes Proof. This is a result of universal-algebraic nonsense: the function assigning to each subalgebra C of Q the subalgebra f- 1 [C] of B is one-one, and it maps {C: P:::; C:::; Q} into {D: A:::; D:::; B}. In case kerf~ A, it maps onto the latter set: the preimage of such aD is f[D], and P:::; f[D] :::; Q. 23.

So, indeed, 0' < (2w)+. Next we claim (4) 0' is a successor ordinal. Suppose not. For each ~ < w let B~ = {cu~ : IB~I 5: 2w since 0' < (2w)+, so IBI 5: 2w. Hence: CJ < 0'}. Let B = fl~

Without loss of generality T has only one root. Set H = {Tit : t E T} U {0, 1}. The conditions for a semigroup algebra are easily verified. B. Interval algebras. Let A = IntAlg L, where L is a linear ordering with first element 0L. Let H = {[0 L, a) : a E L} U { 1}. Again the indicated conditions are easily checked. C. Free algebras. Let A be freely generated by X, and set H = {x E A : xis a finite product of members of X} U {0, 1}. The indicated conditions clearly hold. 2. Special classes 26 It is also useful to note that if A is a semigroup algebra, then so is DupA.

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