J. Donald Monk

Cylindric set algebras

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Some questions about Boolean algebras

Very recently there has been much progress on some fundamental settheoretic problems concerning Boolean algebras. The purpose of this article is to indicate some problems still left open, put in perspective by what has been shown recently. We have made no a t tempt to completely cover the field with these questions, but hope that for the problems ment ioned here the picture we give is fairly complete. To some extent this is a survey of recent set-theoretical results on Boolean algebras. In particular, part of the information we give here answers questions f rom earlier informal versions of this paper and has been included so as to make clear what no longer is an open problem. We are grateful to R. Bonnet , S. Koppelberg, K. Kunen, R. Laver, R. McKenzie, P. Nyikos, S. Shelah and M. Weese for comments on earlier versions of this article.

Provability with finitely many variables

In first-order logic with equality but with finitely many variables, no finite schema suffices to give a sound and complete axiomatization of the universally valid sentences. The proof uses a rather deep result from algebraic logic. The purpose of this note is to give a rather obvious consequence of a recent theorem of J. S. Johnson [5] (whose proof is based on Monk [7]). The consequence has to do with provability in languages which differ from ordinary first-order languages with equality just in having only finitely many individual variables. Such languages have previously been investigated in Henkin [2 ], Henkin-Tarski [3 ], Jaskowski [4], and Pieczkowski [8]. Our result is that it is impossible to write down finitely many schemata which will give a notion of proof that is sound and complete. See below for a more precise formulation; in particular, the notion of schema is made explicit. What makes our result an easy consequence of Johnsons theorem is the fairly general knowledge of certain connections between logically valid sentences in these restricted first-order languages, and equations which hold identically in each representable polyadic equality algebra. The main portion of the note is devoted to an exposition of these connections. Thanks are due to the referee for aid in the formulation of these connections. We employ the usual set-theoretic notation. f*X is the f-image of the set X. co is the set of all natural numbers. We assume throughout that 3 -<a <& (Our languages will have a variables; the case a <2 is considered in Henkin [2].) ?a is the first-order language with equality with the sequence (vi:i<a) of individual variables and with the sequence (Ri:i<4) of nonlogical constants, where Ri is an a-ary relation symbol for each i<co. We treat 7, -, V, and = as primitive logical symbols; V, A/, +*, and 3 are defined in the usual way. An 5a-structure is a structure 21= (A, Ri)i<,, where A 70 and RiCaA for each i<cw. If Received by the editors November 10, 1969 and, in revised form, March 13, 1970. AMS 1969 subject classifications. Primary 0216, 0218; Secondary 0248.

Pseudo-trees and Boolean algebras

We consider Boolean algebras constructed from pseudo-trees in various ways and make comments about related classes of Boolean algebras.

Recursively Enumerable Sets

In this chapter we shall deal in some detail with the set Σ1 of relations (see 5.24). Such relations are called recursively enumerable for reasons which will shortly become clear. The study of recursively enumerable relations is one of the main branches of recursive function theory. They play a large role in logic. In fact, for most theories the set of Godel numbers of theorems is recursively enumerable. Thus many of the concepts introduced in this section will have applications in our discussion of decidable and undecidable theories in Part III. Unless otherwise stated, the functions in this chapter are unary.

ON AN ALGEBRA OF SETS OF FINITE SEQUENCES

The algebras studied in this paper were suggested to the author by William Craig as a possible substitute for cylindric algebras. Both kinds of algebras may be considered as algebraic versions of first-order logic. Cylindric algebras can be introduced as follows. Let Y be a first-order language, and let % be an Y-structure. We assume that Y has a simple infinite sequence v0, vi, . * * of individual variables, and we take as known what it means for a sequence x = of elements of % to satisfy a formula b of Y in A. Let He be the collection of all sequences x

A very rigid Boolean algebra

A Boolean algebra is constructed having only those endomorphisms corresponding to prime ideals, which are present in any BA. The BA constructed is of powerc, has 2c endomorphisms, and is not rigid in Bonnet’s sense.

The spectrum of partitions of a Boolean algebra

Abstract. The main notion dealt with in this article is where A is a Boolean algebra. A partition of 1 is a family ofnonzero pairwise disjoint elements with sum 1. One of the main reasons for interest in this notion is from investigations about maximal almost disjoint families of subsets of sets X, especially X=ω. We begin the paper with a few results about this set-theoretical notion.Some of the main results of the paper are:• (1) If there is a maximal family of size λ≥κ of pairwise almost disjoint subsets of κ each of size κ, then there is a maximal family of size λ of pairwise almost disjoint subsets of κ+ each of size κ.• (2) A characterization of the class of all cardinalities of partitions of 1 in a product in terms of such classes for the factors; and a similar characterization for weak products.• (3) A cardinal number characterization of sets of cardinals with a largest element which are for some BA the set of all cardinalities of partitions of 1 of that BA.• (4) A computation of the set of cardinalities of partitions of 1 in a free product of finite-cofinite algebras.

On General Boundedness and Dominating Cardinals

For cardinals κ, λ,μ we let bκ,λ,μ be the smallest size of a subset B of λμ unbounded in the sense of ≤κ ; that is, such that there is no function f ∈ λμ such that {α < λ : g(α) > f (α)} has size less than κ for all g ∈ B. Similarly for dκ,λ,μ, the general dominating number, which is the smallest size of a subset B of λμ such that for every g ∈ λμ there is an f ∈ B such that the above set has size less than κ . These cardinals are generalizations of the usual ones for κ = λ = μ = ω. When all three are the same regular cardinal, the relationships between them have been completely described by Cummings and Shelah. We also consider some variants of the functions, following van Douwen, in particular the version b ↑ κ,λ,μ of bκ,λ,μ in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) bμ,μ,cfμ ≤ bcfμ,cfμ,cfμ; (2) for λ ≤ μ, b ↑ κ,λ,μ always exists; (3) if cfλ = cfμ < λ ≤ μ, then bcfμ,cfμ,cfμ = b ↑ λ,λ,μ; (4) dω,μ,μ = d1,μ,μ. For background see Section 1 of the paper. Several open problems are stated. 1 Definitions We make the standing assumptions that we have cardinals κ, λ,μ with (1) κ = 1 or κ is infinite, (2) κ ≤ λ, and (3) λ and μ are infinite. Note in particular that we allow for the possibility that λ > μ. The definitions of our functions depend on some quasi orders defined as follows. For f, g ∈ μ we write f ≤κ g iff |{ξ < λ : f (ξ) > g(ξ)}| < κ, f <κ g iff |{ξ < λ : f (ξ) ≥ g(ξ)}| < κ, f ≤ g iff ∀ξ < λ[ f (ξ) ≤ g(ξ)], f < g iff ∀ξ < λ[ f (ξ) < g(ξ)]. The following obvious proposition can be used to fill in some details below. Received April 23, 2003; accepted February 5, 2004; printed October 26, 2004 2000 Mathematics Subject Classification: Primary, 03E10; Secondary, 03E35

Minimum‐sized Infinite Partitions of Boolean Algebras

For any Boolean Algebra A, let cmm(A) be the smallest size of an infinite partition of unity in A. The relationship of this function to the 21 common functions described in Monk [4] is described, for the class of all Boolean algebras, and also for its most important subclasses. This description involves three main results: the existence of a rigid tree algebra in which cmm exceeds any preassigned number, a rigid interval algebra with that property, and the construction of an interval algebra in which every well-ordered chain has size less than cmm. Mathematics Subject Classification: 06E05, 03E05.