Atoms in the lattice of covering operators in compact Hausdorff spaces
Section snippets
Introduction and preliminaries
A minimum proper cover (mpc) is a proper cover such that every proper cover of X is a cover of Y. The main theorems of this paper are: First (§2), the characterization of the mpc's as the covers with Y infinite and extremally disconnected and f identifying only two different non-isolated points of Y. Second (§3), the atoms in the lattice coComp correspond one-to-one with the mpc's. Third (§3), every element of coComp different from the identity operator lies above an atom. Finally (§4
The mpc's
Before the theorem here (2.2), we review our situation.
Definitions, etc. 2.1 (a) Let L be a lattice (not necessarily a set) with a bottom 0. An atom in L is a minimal element of . A strong atom in L (we coin the term) is an atom in L which is minimum in (thus unique). Birkhoff [4, p. 194] defines an element y of a lattice L (not assuming L has a bottom) to be strictly meet irreducible just in case has a least element, say . Thus, if L has 0, then a is the strong atom if and only if 0 is
mpc's and atoms in coComp
Recall (1.1(c)) that coComp is a complete lattice.
Theorem 3.1 If is an mpc, then an atom in coComp is defined for by (Note: .) If a is an atom in coComp, then there is an mpc γ with .
Proof (a) We show first that via the equivalence given in 1.1(c). Let . We show that . If , then by the definition of , so and thus . So suppose . Since , one
coComp is not atomic
We shall establish another fact about the lattice coComp, made possible by the characterization in Theorem 3.1.
Theorem 4.1 The Gleason covering operator is not the supremum of the atoms of coComp. The lattice coComp is not atomic.
We are denoting an mpc γ as . Recall that for , if and only if .
Lemma 4.2 Suppose . Let the supremum in the complete lattice coComp. Then if and only if there exists . The latter condition
Final remarks
“Covering theory” as described in Comp here, can be described in general categories and, as such, has an opposite/dual theory, which is that of essential extensions; the dual of a covering operator is an essential extension operator, frequently called a hull operator (ho). See [2] and [1, §9] for discussions of this duality. It remains a question that, with appropriate assumptions about the category C, to what extent the dual of Theorem 3.1 carries over, giving the same correspondence between
Acknowledgements
We thank the referee for several helpful comments.
We are profoundly in the debt of Bernhard Banaschewski, who identified a gap in our original proof of 3.1(b), and was kind enough to discuss extensively the situation with us.
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