By J. Donald Monk

ISBN-10: 3034807295

ISBN-13: 9783034807296

ISBN-10: 3034807309

ISBN-13: 9783034807302

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

Assuming familiarity with in simple terms the fundamentals of Boolean algebras and set thought, via easy limitless combinatorics and forcing, the e-book reports present wisdom approximately those services, giving entire proofs for many evidence. a different function of the booklet 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 a similar writer, the current paintings is way higher than both of those. It comprises ideas to a few of the open difficulties of the sooner volumes. one of the new subject matters are continuum cardinals on Boolean algebras, with a long remedy of the reaping quantity. Diagrams on the finish of the booklet summarize the relationships among the features for plenty of very important periods of Boolean algebras, together with period algebras, tree algebras and superatomic algebras.

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

**Example text**

S(am−1 )), m > 0. Since G ∈ yΓ , some ai is coﬁnite. By the above, if {ai : ai ﬁnite } ⊆ Δ ⊆ a0 ∪ · · · ∪ am−1 , then yΔ ∈ V (S(a0 ), . . , S(am−1 )). Thus V (S(a0 ), . . , S(am−1 )) has inﬁnitely many members, as desired. We also proved: (5) {yΓ : Γ ⊆ κ}, as a subspace of Ult(A), is closed and has no isolated points. The following is obvious: (6) {{zΓ } : Γ ﬁnite and non-empty} is the set of all atoms of Exp(A), which is atomic. Let xα = V (S(κ\{α}), S({α})) for all α < κ. (7) xα /ﬁn : α < κ is a system of independent elements of Exp(A)/ﬁn.

Special Classes of Boolean Algebras 43 Now we want to give an abstract characterization of pseudo-tree algebras. For this purpose we need some easy propositions. A subset R of a BA A is a ramiﬁcation set provided that any two elements of R are either comparable or disjoint. Thus R is then a pseudo-tree under the inverse ordering of the BA. 11. Let (P, ≤) be a partially ordered system. Then {P ↑ p : p ∈ P } is a disjunctive set in P(P ). Proof. Obviously ∅ is not in the indicated set. Now suppose that p, p1 , .

Thus t∈H (T ↑ t) ∩ s∈G [T \(T ↑ s)] ∈ F , so F ∈S (T ↑ t) ∩ t∈H [T \(T ↑ s)] ⊆ U. s∈G Now we give a theorem which is deeper than the ones presented so far. The hard part, (ii)⇒(i), is due to Purisch and Nikiel, who proved it topologically. The present purely algebraic proof is due to Heindorf [97]. For the easy part we need two simple lemmas. 19. If T is a pseudo-tree and t ∈ T , then (Treealg(T )) same as Treealg(T ↑ t). 20. If T is a ﬁnite tree with a least element r, then Treealg(T ) can be embedded in an interval algebra.

### Cardinal Invariants on Boolean Algebras: Second Revised Edition by J. Donald Monk

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