Description:
<p>A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa">
<mml:semantics>
<mml:mi>κ<!-- κ --></mml:mi>
<mml:annotation encoding="application/x-tex">\kappa</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> there are <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript kappa">
<mml:semantics>
<mml:mrow class="MJX-TeXAtom-ORD">
<mml:msup>
<mml:mn>2</mml:mn>
<mml:mi>κ<!-- κ --></mml:mi>
</mml:msup>
</mml:mrow>
<mml:annotation encoding="application/x-tex">{2^\kappa }</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> isomorphism types of Boolean algebras of power <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa">
<mml:semantics>
<mml:mi>κ<!-- κ --></mml:mi>
<mml:annotation encoding="application/x-tex">\kappa</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa">
<mml:semantics>
<mml:mi>κ<!-- κ --></mml:mi>
<mml:annotation encoding="application/x-tex">\kappa</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-distributive complete Boolean algebra can be completely embedded in a <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="kappa">
<mml:semantics>
<mml:mi>κ<!-- κ --></mml:mi>
<mml:annotation encoding="application/x-tex">\kappa</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-distributive complete Boolean algebra with no rigid or homogeneous factors.</p>