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total order
Chain-meet-closed sets ★★
Author(s): Porton
Let is a complete lattice. I will call a filter base a nonempty subset
of
such that
.
Definition A subset
of a complete lattice
is chain-meet-closed iff for every non-empty chain
we have
.
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ T\in\mathscr{P}S $](/files/tex/57e116e15e721bcdb86104a2c28f1cdfb0e6df9e.png)
![$ \bigcap T\in S $](/files/tex/8f696ee668ef8dd7ea6c8a64b9669645285fc295.png)
Conjecture A subset
of a complete lattice
is chain-meet-closed iff for every filter base
we have
.
![$ S $](/files/tex/d2b76a0ee5465d3e3ecc846c8e3d632edd8b2bbf.png)
![$ \mathfrak{A} $](/files/tex/702b76abd81b24daaf0e6bc2a191fb964b09d1b2.png)
![$ T\in\mathscr{P}S $](/files/tex/57e116e15e721bcdb86104a2c28f1cdfb0e6df9e.png)
![$ \bigcap T\in S $](/files/tex/8f696ee668ef8dd7ea6c8a64b9669645285fc295.png)
Keywords: chain; complete lattice; filter bases; filters; linear order; total order
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