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Cellularity of free products of Boolean algebras (or topologies)

Saharon Shelah (2000)

Fundamenta Mathematicae

The aim this paper is to present an answer to Problem 1 of Monk [10], [11]. We do this by proving in particular that if μ is a strong limit singular cardinal, θ = ( 2 c f ( μ ) ) + and 2 μ = μ + then there are Boolean algebras 𝔹 1 , 𝔹 2 such that c ( 𝔹 1 ) = μ , c ( 𝔹 2 ) < θ b u t c ( 𝔹 1 * 𝔹 2 ) = μ + . Further we improve this result, deal with the method and the necessity of the assumptions. In particular we prove that if 𝔹 is a ccc Boolean algebra and μ ω λ = c f ( λ ) 2 μ then 𝔹 satisfies the λ-Knaster condition (using the “revised GCH theorem”).

Certain partial orders on semigroups

Mario Petrich (2001)

Czechoslovak Mathematical Journal

Relations introduced by Conrad, Drazin, Hartwig, Mitsch and Nambooripad are discussed on general, regular, completely semisimple and completely regular semigroups. Special properties of these relations as well as possible coincidence of some of them are investigated in some detail. The properties considered are mainly those of being a partial order or compatibility with multiplication. Coincidences of some of these relations are studied mainly on regular and completely regular semigroups.

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