Boolean algebras with no rigid or homogeneous factors

Štěpánek, Petr

doi:10.1090/S0002-9947-1982-0642333-3

Boolean algebras with no rigid or homogeneous factors
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by Petr Štěpánek

Trans. Amer. Math. Soc. 270 (1982), 131-147

DOI: https://doi.org/10.1090/S0002-9947-1982-0642333-3

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Abstract:

A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal $\kappa$ there are ${2^\kappa }$ isomorphism types of Boolean algebras of power $\kappa$ 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 $\kappa$-distributive complete Boolean algebra can be completely embedded in a $\kappa$-distributive complete Boolean algebra with no rigid or homogeneous factors.

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