A note on infinite partitions of free products of Boolean algebras

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Publication date: 27 February 2022

Abstract: If

A

is an infinite Boolean algebra the cardinal invariant

m a t h f r a k a (A)

is defined as the smallest size of an infinite partition of

A

. The cardinal

m a t h f r a k a (A o p l u s B)

, where

A o p l u s B

is the free product of the Boolean algebras

A

and

B

(whose dual topological space is the product of the dual topological spaces of

A

and

B

), is below both

m a t h f r a k a (A)

and

m a t h f r a k a (B)

. The equality

m a t h f r a k a (A o p l u s B) = m i n l b r a c e m a t h f r a k a (A), m a t h f r a k a (B) b r a c e

is not known to hold for all infinite Boolean algebras

A

and

B

. Here some lower bounds of

m a t h f r a k a (A o p l u s B)

are provided.

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