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So I know that there exists an infinite Boolean algebra that has atoms, for example the power set on the natural numbers has an infinite number of atoms, ie the singleton sets.

But is there a Boolean algebra (infinite) which only has exactly one atom? Or would this imply some sort of contradiction? Or is there a trivial/canonical example of such?

Thanks

Quality
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1 Answers1

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Let $B$ be any boolean algebra without atoms, such as the free boolean algebra on countably infinitely many generators. Let $B'$ be the algebra $2$. Then $B \times B'$ has precisely one atom, namely the element $(0,1)$.

Mees de Vries
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  • Thanks, this is only a first course in logic for me. I am not sure what you mean by the algebra 2? Do you mean the algebra generated by the Power set of 2? – Quality Oct 23 '16 at 00:58
  • @Quality, 2 is the unique 2-element Boolean algebra, i.e., the algebra consisting only of a top and a bottom element, which are distinct. – Mees de Vries Oct 23 '16 at 08:02