View source for Parovichenko algebra

''Parovičenko algebra''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200601.png" /> be a [[Boolean algebra|Boolean algebra]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200602.png" />, then one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200603.png" /> provided that for all finite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200604.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200605.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200606.png" />. In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200607.png" /> is said to be a Parovichenko algebra provided that it is both Cantor- and DuBois-Reymond-separable. This means that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p1200609.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006010.png" /> there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006011.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006012.png" />.

The Parovichenko theorem from 1963 (see [[#References|[a3]]]) asserts that under the [[Continuum hypothesis|continuum hypothesis]] (abbreviated CH), every Parovichenko algebra of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006013.png" /> (i.e., of the cardinality of the continuum, cf. also [[Continuum, cardinality of the|Continuum, cardinality of the]]) is isomorphic to the Boolean algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006015.png" /> is the power set algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006017.png" /> is its ideal of finite subsets. The theorem is proved by a [[Transfinite induction|transfinite induction]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006018.png" /> many steps, where at each essential step of the process the separability properties of the Boolean algebras under consideration ensure that one can continue with the construction. This method goes back to W. Rudin [[#References|[a7]]].

In 1978, E.K. van Douwen and J. van Mill proved the converse to Parovichenko's theorem in [[#References|[a1]]]: If all Parovichenko algebras of size of the continuum are isomorphic, then the continuum hypothesis holds.

Parovichenko's theorem implies that every Boolean algebra of size at most that of the continuum can be embedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006019.png" /> under the continuum hypothesis. This result cannot be proved in ZFC alone (cf. also [[Set theory|Set theory]]). In 1968, K. Kunen [[#References|[a3]]] proved that in a model formed by adding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006020.png" /> Cohen reals to a model of the continuum hypothesis, there is no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006021.png" /> sequence of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006022.png" /> which is strictly decreasing (modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006023.png" />). This implies that a Boolean algebra such as the clopen algebra of the ordinal space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006024.png" /> cannot be embedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006025.png" />. In 1978, E.K. van Douwen and T.C. Przymusiński [[#References|[a2]]] used results of F. Rothberger [[#References|[a6]]] to prove that there is a counterexample under the hypothesis

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006026.png" /></td> </tr></table>

This is interesting, since it only involves a hypothesis on cardinal numbers.

Parovichenko's theorem has interesting consequences in topology. It implies, for example, that under the continuum hypothesis the Čech–Stone remainder <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006027.png" /> of any zero-dimensional locally compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006028.png" />-compact, non-compact space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006029.png" /> of weight at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006030.png" /> is homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006031.png" />, the Čech–Stone-remainder of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006032.png" /> with the discrete topology (cf. also [[Stone–Čech compactification|Stone–Čech compactification]]; [[Topological structure (topology)|Topological structure (topology)]]; [[Zero-dimensional space|Zero-dimensional space]]).

For more information, see e.g. [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.K. van Douwen,   J. van Mill,   "Parovičenko's characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006033.png" /> implies CH"  ''Proc. Amer. Math. Soc.'' , '''72'''  (1978)  pp. 539–541</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.K. van Douwen,   T.C. Przymusiński,   "Separable extensions of first countable spaces"  ''Fund. Math.'' , '''105'''  (1980)  pp. 147—158</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Kunen,   "Inaccessibility properties of cardinals"  ''PhD Thesis Stanford Univ.''  (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. van Mill,   "An introduction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006034.png" />"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set–Theoretic Topology'' , North-Holland  (1984)  pp. 503–567</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  I.I. Parovičenko,   "A universal bicompact of weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120060/p12006035.png" />"  ''Soviet Math. Dokl.'' , '''4'''  (1963)  pp. 592–592</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F. Rothberger,   "A remark on the existence of a denumberable base for a family of functions"  ''Canad. J. Math.'' , '''4'''  (1952)  pp. 117–119</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  W. Rudin,   "Homogeneity problems in the theory of Čech compactifications"  ''Duke Math. J.'' , '''23'''  (1956)  pp. 409–419</TD></TR></table>