On the lattice of weak topologies on the bicyclic monoid with adjoined zero
Keyword(s):
A Hausdorff topology τ on the bicyclic monoid with adjoined zero C0 is called weak if it is contained in the coarsest inverse semigroup topology on C0. We show that the lattice W of all weak shift-continuous topologies on C0 is isomorphic to the lattice SIF1×SIF1 where SIF1 is the set of all shift-invariant filters on ω with an attached element 1 endowed with the following partial order: F≤G if and only if G=1 or F⊂G. Also, we investigate cardinal characteristics of the lattice W. In particular, we prove that W contains an antichain of cardinality 2c and a well-ordered chain of cardinality c. Moreover, there exists a well-ordered chain of first-countable weak topologies of order type t.
1978 ◽
Vol 19
(1)
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pp. 59-65
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2010 ◽
Vol 81
(2)
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pp. 195-207
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2014 ◽
Vol 90
(1)
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pp. 121-133
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1980 ◽
Vol 23
(3)
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pp. 249-260
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2006 ◽
Vol 81
(2)
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pp. 185-198
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Keyword(s):
1989 ◽
Vol 47
(3)
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pp. 399-417
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