By J. Donald Monk

This ebook is anxious with cardinal quantity valued features outlined for any Boolean algebra. Examples of such capabilities are independence, which assigns to every Boolean algebra the supremum of the cardinalities of its unfastened subalgebras, and cellularity, which supplies the supremum of cardinalities of units of pairwise disjoint parts. Twenty-one such services are studied intimately, and lots of extra in passing. The questions thought of are the behaviour of those services lower than algebraic operations reminiscent of items, loose items, ultraproducts, and their relationships to 1 another.

Assuming familiarity with merely the fundamentals of Boolean algebras and set idea, via basic countless combinatorics and forcing, the ebook studies present wisdom approximately those features, giving whole proofs for many proof. a distinct function of the e-book is the eye given to open difficulties, of which 185 are formulated.

Based on *Cardinal capabilities on Boolean Algebras* (1990) and *Cardinal Invariants on Boolean Algebras* (1996) by way of an identical writer, the current paintings is way higher than both of those. It comprises suggestions to the various open difficulties of the sooner volumes. one of the new subject matters are continuum cardinals on Boolean algebras, with a long therapy of the reaping quantity. Diagrams on the finish of the e-book summarize the relationships among the capabilities for lots of vital sessions of Boolean algebras, together with period algebras, tree algebras and superatomic algebras.

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**Extra resources for Cardinal Invariants on Boolean Algebras: Second Revised Edition**

**Example text**

Set H = {T ↑ t : t ∈ T } ∪ {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 = {[0L , a) : a ∈ L} ∪ {1}. Again the indicated conditions are easily checked. C. Free algebras. Let A be freely generated by X, and set H = {x ∈ A : x is a ﬁnite product of members of X} ∪ {0, 1}. The indicated conditions clearly hold. It is also useful to note that if A is a semigroup algebra, then so is Dup(A).

Write x = a + b, with a ∈ A x and b ∈ A −x. Clearly b = 0, so x x = a ∈ A. 25. Let A(x) be a simple extension of A, and let a ∈ A. Then the following conditions are equivalent: a ∈ SmpA x; a = b + c for some b ∈ A x and c ∈ A −x; a · x ∈ A; a · −x ∈ A; For all y ∈ A(x), if y ≤ a then y · x ∈ A and y · −x ∈ A, and so y ∈ A and A(x) a = A a; (vi) For all y ∈ A(x), if y ≤ a then y ∈ SmpA x. (i) (ii) (iii) (iv) (v) Proof. Clearly (i)⇔(ii). Assume (ii). , (iii) holds. Assume (iii). , (iv) holds. Similarly (iv)⇒(iii).

Thus (iii) holds. For (ii), suppose that u < v. If u = r, say si ≤ v. Then g(T ↑ u) = [t0 , ∞) and g(T ↑ v) = fi (T ↑ v) = [cv , dv ), and t0 < cv and dv ≤ ti+1 < ∞. If u = r, the conclusion is clear. So (ii) holds. For (d), suppose that w ∈ T , F is a ﬁnite subset of T , and ∀x ∈ F [x ≤ w]. Case 1. w = r. Now clearly for all x ∈ F we have r ∈ / g(T ↑ x). Hence w ∈ x∈F [M \g(T ↑ x)], as desired. Case 2. r < w. Say si ≤ w. Now take any x ∈ F . Thus g(T ↑ x) = fi (T ↑ x) = [cx , dx ). We claim that cw ∈ / g(T ↑ x).