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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Structure diagrams for primitive Boolean algebras


Author: James Williams
Journal: Proc. Amer. Math. Soc. 47 (1975), 1-9
MathSciNet review: 0357269
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Abstract | References | Additional Information

Abstract: If $ S$ and $ T$ are structure diagrams for primitive Boolean algebras, call a homomorphism $ f$ from $ S$ onto $ T$ right-strong iff whenever $ xTf(t)$, there is an $ s$ such that $ f(s) = x$ and $ sSt$; let RSE denote the category of diagrams and onto right-strong homomorphisms. The relation ``$ S$ structures $ \mathfrak{B}$'' between diagrams and Boolean algebras induces a 1-1 correspondence between the components of RSE and the isomorphism types of primitive Boolean algebras. Up to isomorphism, each component of RSE contains a unique minimal diagram and a unique maximal tree diagram. The minimal diagrams are like those given in a construction by William Hanf. The construction which is given for producing maximal tree diagrams is recursive; as a result, every diagram $ S$ structures a Boolean algebra recursive in $ S$.


References [Enhancements On Off] (What's this?)

  • [1] William Hanf, Primitive Boolean algebras, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971) Amer. Math. Soc., Providence, R.I., 1974, pp. 75–90. MR 0379182 (52 #88)
  • [2] R. S. Pierce, Compact zero-dimensional metric spaces of finite type, American Mathematical Society, Providence, R.I., 1972. Memoirs of the American Mathematical Society, No. 130. MR 0357268 (50 #9736)


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0357269-7
PII: S 0002-9939(1975)0357269-7
Article copyright: © Copyright 1975 American Mathematical Society