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For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B1 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. ■ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such that $B\upharpoonright a$ is an atomic Boolean algebra and $B\upharpoonright s$ is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that $\langle A, \leq\rangle$ is a partial well-quasi-ordering, if it is a partial quasi-ordering and for every $\{a_i\mid i \in \omega\} \subseteq A$ , there are $i < j < \omega$ such that ai ≤ aj. Theorem 2. $\langle M_{T_\omega}, \leq\rangle$ contains a subset M such that the partial orderings $\langle M, \leq \upharpoonright M \rangle$ and $\langle\mathscr{P}(\omega), \subseteq\rangle$ are isomorphic. ■ Let M'0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M'0, let B1 ≤' B2 mean that B1 is embeddable in B2. Remark. $\langle M'_0, \leq'\rangle$ is well-quasi-ordered. ■ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering.

Let B be a superatomic Boolean algebra (BA). The rank of B (rk(B)), is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B - {0}, then the rank of a in B (rk(a)), is defined to be the rank of the Boolean algebra $B b \upharpoonright a \overset{\mathrm{def}}{=} \{b \in B: b \leq a\}$ . The rank of 0 B is defined to be -1. An element a ∈ B - {0} is a generalized atom $(a \in \widehat{At}(B))$ , if the last nonzero cardinal in the cardinal sequence of B $\upharpoonright$ a is 1. Let a,b $\in\widehat{At}$ (B). We denote a ∼ b, if rk(a) = rk(b) = rk(a · b). A subset H $\subseteq \widehat{At}$ (B) is a complete set of representatives (CSR) for B, if for every a $\in \widehat{At}$ (B) there is a unique h ∈ H such that h ∼ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B. THEOREM 1. Let B be a Boolean algebra with cardinal sequence $\langle\aleph_0: i . If B is CWG, then every subalgebra of B is CWG. A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1. Theorem 1 follows from Theorem 2.9, which is the main result of this work. For an ESL BA B we define a set F B of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent. (1) Every subalgebra of B is CWG; and (2) F B is bounded. THEOREM 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated.

IN THEIR WELL-KNOWN PAPER, Kochen and Specker (1967) introduce the concept of partial Boolean algebra (pBa) and show that certain (finitely generated) partial Boolean algebras arising in quantum theory fail to possess morphisms to any Boolean algebra (we call such pBa's intractable in the sequel). In this note we begin by discussing partial..

In this note we shall describe the lattice of the congruences on a balanced Ockham algebra with the pseudocomplementation whose quotient algebras are boolean. This is an extension of the result obtained by Rodrigues and Silva who gave a description of the lattice of congruences on an Ockham algebra whose quotient algebras are boolean.

The notion of monadic three-valued ukasiewicz algebras was introduced by L. Monteiro ([12], [14]) as a generalization of monadic Boolean algebras. A. Monteiro ([9], [10]) and later L. Monteiro and L. Gonzalez Coppola [17] obtained a method for the construction of a three-valued ukasiewicz algebra from a monadic Boolea algebra. In this note we give the construction of a monadic three-valued ukasiewicz algebra from a Boolean algebra B where we have defined two quantification operations and * such that *x=*x (where *x=-*-x). In this case we shall say that and * commutes. If B is finite and is an existential quantifier over B, we shall show how to obtain all the existential quantifiers * which commute with .Taking into account R. Mayet [3] we also construct a monadic three-valued ukasiewicz algebra from a monadic Boolean algebra B and a monadic ideal I of B.

De Morgan's Formal Logic, which was published on virtually the same day in 1847 as Boole's The Mathematical Analysis of Logic, contains a logic of complex terms (LCT) which has been sadly neglected. It is surprising to find that LCT contains almost a full theory of Boolean algebra. This paper will: (1) provide some background to LCT; (2) outline its main features; (3) point out some gaps in it; (4) compare it with Boole's algebra; (5) show that it is a lattice-theoretical formulation of Boolean algebra; (6) discuss some issues of historical priority; and (7) conclude with the puzzle of LCT's lack of influence.

A Boolean algebra B is said to be openly generated if {A: A ≤rc B, |A| = ℵ0} includes a club subset of [ B]ℵ0 . We show: (V = L). For any cardinal κ there exists an L∞κ-free Boolean algebra which is not openly generated (Proposition 4.1). (MA+(σ-closed)). Every L∞ℵa -free Boolean algebra is openly generated (Theorem 4.2). The last assertion follows from a characterization of openly generated Boolean algebras under MA+(σ-closed) (Theorem 3.1). Using this characterization we also prove the independence of problem 7 in Scepin [15] (Proposition 4.3 and Theorem 4.4).

We show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebra B has an irredundant subset A such that 2 |A| = 2 |B| . This implies in particular that B has 2 |B| subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra. The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a "black box" at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.

We show that there is a computable Boolean algebra B and a computably enumerable ideal I of B such that the quotient algebra B/I is of Cantor-Bendixson rank 1 and is not isomorphic to any computable Boolean algebra. This extends a result of L. Feiner and is deduced from Feiner's result even though Feiner's construction yields a Boolean algebra of infinite Cantor-Bendixson rank.

The standard model for mereotopological structures are Boolean subalgebras of the complete Boolean algebra of regular closed subsets of a nonempty connected regular T0 topological space with an additional "contact relation" C defined by xCy ? x n ? Ø.