Displaying similar documents to “Prime ideal theorem for double Boolean algebras”

On BPI Restricted to Boolean Algebras of Size Continuum

Eric Hall, Kyriakos Keremedis (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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(i) The statement P(ω) = “every partition of ℝ has size ≤ |ℝ|” is equivalent to the proposition R(ω) = “for every subspace Y of the Tychonoff product 2 ( ω ) the restriction |Y = Y ∩ B: B ∈ of the standard clopen base of 2 ( ω ) to Y has size ≤ |(ω)|”. (ii) In ZF, P(ω) does not imply “every partition of (ω) has a choice set”. (iii) Under P(ω) the following two statements are equivalent: (a) For every Boolean algebra of size ≤ |ℝ| every filter can be extended to an ultrafilter. (b) Every Boolean...

Ideal independence, free sequences, and the ultrafilter number

Kevin Selker (2015)

Commentationes Mathematicae Universitatis Carolinae

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We make use of a forcing technique for extending Boolean algebras. The same type of forcing was employed in Baumgartner J.E., Komjáth P., Boolean algebras in which every chain and antichain is countable, Fund. Math. 111 (1981), 125–133, Koszmider P., Forcing minimal extensions of Boolean algebras, Trans. Amer. Math. Soc. 351 (1999), no. 8, 3073–3117, and elsewhere. Using and modifying a lemma of Koszmider, and using CH, we obtain an atomless BA, A such that 𝔣 ( A ) = s mm ( A ) < 𝔲 ( A ) , answering questions raised...

Tychonoff Products of Two-Element Sets and Some Weakenings of the Boolean Prime Ideal Theorem

Kyriakos Keremedis (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let X be an infinite set, and (X) the Boolean algebra of subsets of X. We consider the following statements: BPI(X): Every proper filter of (X) can be extended to an ultrafilter. UF(X): (X) has a free ultrafilter. We will show in ZF (i.e., Zermelo-Fraenkel set theory without the Axiom of Choice) that the following four statements are equivalent: (i) BPI(ω). (ii) The Tychonoff product 2 ℝ , where 2 is the discrete space 0,1, is compact. (iii) The Tychonoff product [ 0 , 1 ] ℝ is compact. (iv) In a...

Topological representation for monadic implication algebras

Abad Manuel, Cimadamore Cecilia, Díaz Varela José (2009)

Open Mathematics

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In this paper, every monadic implication algebra is represented as a union of a unique family of monadic filters of a suitable monadic Boolean algebra. Inspired by this representation, we introduce the notion of a monadic implication space, we give a topological representation for monadic implication algebras and we prove a dual equivalence between the category of monadic implication algebras and the category of monadic implication spaces.

Functional monadic n -valued Łukasiewicz algebras

A. V. Figallo, Claudia A. Sanza, Alicia Ziliani (2005)

Mathematica Bohemica

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Some functional representation theorems for monadic n -valued Łukasiewicz algebras (qLk n -algebras, for short) are given. Bearing in mind some of the results established by G. Georgescu and C. Vraciu (Algebre Boole monadice si algebre Łukasiewicz monadice, Studii Cercet. Mat. 23 (1971), 1027–1048) and P. Halmos (Algebraic Logic, Chelsea, New York, 1962), two functional representation theorems for qLk n -algebras are obtained. Besides, rich qLk n -algebras are introduced and characterized....

On the injectivity of Boolean algebras

Bernhard Banaschewski (1993)

Commentationes Mathematicae Universitatis Carolinae

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The functor taking global elements of Boolean algebras in the topos 𝐒𝐡 𝔅 of sheaves on a complete Boolean algebra 𝔅 is shown to preserve and reflect injectivity as well as completeness. This is then used to derive a result of Bell on the Boolean Ultrafilter Theorem in 𝔅 -valued set theory and to prove that (i) the category of complete Boolean algebras and complete homomorphisms has no non-trivial injectives, and (ii) the category of frames has no absolute retracts.