# Cardinal Invariants on Boolean Algebras by J. Donald Monk

By J. Donald Monk

This e-book is worried with cardinal quantity valued features outlined for any Boolean algebra. Examples of such features are independence, which assigns to every Boolean algebra the supremum of the cardinalities of its loose subalgebras, and cellularity, which provides the supremum of cardinalities of units of pairwise disjoint parts. Twenty-one such features are studied intimately, and lots of extra in passing. The questions thought of are the behaviour of those capabilities lower than algebraic operations akin to items, loose items, ultraproducts, and their relationships to each other. Assuming familiarity with simply the fundamentals of Boolean algebras and set idea, via to easy countless combinatorics and forcing, the ebook stories present wisdom approximately those features, giving entire proofs for many evidence. a unique function of the e-book is the eye given to open difficulties, of which ninety seven are formulated.

Based on Cardinal features on Boolean Algebras (1990) by way of an analogous writer, the current paintings is almost two times the scale of the unique paintings. It includes recommendations to a few of the open difficulties that are mentioned in larger element than prior to. one of the new issues thought of are ultraproducts and Fedorchuk?s theorem, and there's a extra entire remedy of the cellularity of loose items. Diagrams on the finish of the publication summarize the relationships among the features for lots of very important periods of Boolean algebras, together with tree algebras and superatomic algebras.

Review:

"This ebook is an necessary instrument for somebody operating in Boolean algebra, and is additionally advised for set-theoretic topologists."- Zentralblatt MATH

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By the above lemmas, it suﬃces to work with partial orders. A function λ is said to satisfy condition (α) provided that λ is an ω-termed sequence of inﬁnite cardinals such that supξ<ω λξ = ω and λη < λξ = cfλξ for all ξ < ω. η<ξ For the following deﬁnitions, assume that λ satisﬁes condition (α). For distinct a, b ∈ ξ<ω λξ let ρ(a, b) = min{ξ < ω : aξ = bξ}. For a, b ∈ ξ<ω λξ we deﬁne a =∗ a ≤∗ a ∗ a <∗ b b b b iff iff iff iff ∃ξ < ω∀η ∈ (ξ, ω)[aη = bη]; ∃ξ < ω∀η ∈ (ξ, ω)[aη ≤ bη]; a ≤∗ b and a =∗ b; ∃ξ < ω∀η ∈ (ξ, ω)[aη < bη].

Simple extensions of Boolean algebras Given BA’s A and B, we call B a simple extension of A provided that A is a subalgebra of B and B = A ∪ {x} for some x ∈ B; then we write B = A(x). We recall that each element of A(x) can be written in the form a · x + b · −x with a, b ∈ A; or in the form c + a · x + b · −x with a, b, c pairwise disjoint elements of A. We now introduce some important ideals for studying simple extensions. If A is a subalgebra of B and x ∈ B, we let A x = {a ∈ A : a ≤ x}. This is a slight extension of the usual notion; x is not necessarily in A.

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 a D is f [D], and P ≤ f [D] ≤ Q. 23. Assume that A ≤ B ≤ M ≥ D. Set P = A∩D and Q = B∩D. Then A ≤m B implies that P ≤m Q. Proof. Assume all the hypotheses. We may also assume that P = Q.