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Annali di Matematica Pura ed Applicata

, Volume 126, Issue 1, pp 253–266 | Cite as

Lo spazio duale di un prodotto di algebre di Boole e le compattificazioni di Stone

  • Claudio Bernardi
Article

Summary

This paper is concerned with the Stone space X of a direct product\(B = \prod\limits_{i \in I} {B_i }\) of infinitely many Boolean algebras. In paragraph 2, after recalling that X is the Stone-Čech compactification of the sum (disjoint union)\(\sum\limits_{i \in I} {X_i }\) of the Stone spaces of the algebras Bi, we exhibit a compactification of\(\sum\limits_{i \in I} {X_i }\) which is not a Stone space and we give a method to construct all the «Stone compactifications» of\(\sum\limits_{i \in I} {X_i }\) (the corresponding Boolean algebras are easily characterised). In paragraph 3, a set of ultrafilters of B (the «decomposable» ultrafilters) are introduced: this set properly contains\(\sum\limits_{i \in I} {X_i }\), but, as is shown in paragraph 5, there are direct products that admit nondeeomposable ultrafilters (this is the case iff the set {Card Bi: i ε I } is not bounded by a natural number). In paragraph 4, among other things, we prove, for the set of decomposable ultrafilters, a weak form of countable compactness, in the sense that every countable clopen cover has a finite subcover; then, we deduce that the set of decomposable ultrafilters is pseudocompact, while obviously\(\sum\limits_{i \in I} {X_i }\) is not. Lastly, in paragraph 6, we give a second characterisation of the Stone space of B, showing that every ultrafilter of B can be obtained by iterating in a suitable way the procedure which leads to the construction of decomposable ultrafilters.

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Bibliografia

  1. [1]
    R. E. Chandler,Hausdorff compactifications, Marcel Dekker, New York, 1976.Google Scholar
  2. [2]
    W. W. Comfort -S. Negrepontis,The Theory of Ultrafilters, Springer-Verlag, Berlino, 1974.Google Scholar
  3. [3]
    Ph. Dwinger,Remarks on the field representation of Boolean algebras, Indag. Math.,22 (1960), pp. 213–217.Google Scholar
  4. [4]
    Z. Frolik,Sums of ultrafilters, Bull. Amer. Math. Soc.,73 (1967), pp. 87–91.Google Scholar
  5. [5]
    P. R. Halmos,Algebraic Logic I:Monadic Boolean Algebras, Compositio Math.,12 (1955), pp. 217–249 (ristampato inAlgebraic Logic, Chelsea Publ. Co., New York, 1962).Google Scholar
  6. [6]
    P. R. Halmos,Lectures on Boolean Algebras, Van Nostrand, Toronto, 1963.Google Scholar
  7. [7]
    R. Magari,Varietà a quozienti filtrali, Ann. Un. Ferrara (Nuova Serie), Sez. VII,14 (1969), pp. 5–20.Google Scholar
  8. [8]
    R.Magari,Congruenze di un prodotto diretto legate alle congruenze dei fattori (Congruenze ideali I) (Algebre a congruenze speciali, parte III), in Atti del Convegno di Teoria dei Modelli, Roma, 1969.Google Scholar
  9. [9]
    R. Magari,Representation and duality theory for diagonalizable algebras, Studia Logica,34 (1975), pp. 305–313.Google Scholar
  10. [10]
    R. S. Pierce,Rings of integer-valued continuous functions, Trans. Amer. Math. Soc.,100 (1961), pp. 371–394.Google Scholar
  11. [11]
    R. Sikorski,Cartesian product of Boolean algebras, Fund. Math.,37 (1950), pp. 25–54.Google Scholar
  12. [12]
    R. Sikorski,Boolean Algebras, Springer, Berlino, 1964.Google Scholar
  13. [13]
    S. Stefani,On the representation of Hemimorphisms between Boolean Algebras, Boll. Un. Mat. Ital.,13-A (1976), pp. 206–211.Google Scholar
  14. [14]
    J. Van der Slot,A survey of realcompactness, inTheory of Sets and Topology, curato da Asser et al., VEB Deutscher der Wissenschaften, Berlino, 1972, pp. 473–494.Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1980

Authors and Affiliations

  • Claudio Bernardi
    • 1
  1. 1.Siena

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