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Coatoms of the lattice of funcoids (Solved)

Importance: Low ✭
Author(s): Porton, Victor
Subject: Topology
Keywords: atoms
coatoms
funcoids
Recomm. for undergrads: no
Posted by: porton
on: April 24th, 2014
Solved by: Porton, Victor
Problem   Let $ A $ and $ B $ be infinite sets. Characterize the set of all coatoms of the lattice $ \mathsf{FCD}(A;B) $ of funcoids from $ A $ to $ B $. Particularly, is this set empty? Is $ \mathsf{FCD}(A;B) $ a coatomic lattice? coatomistic lattice?

See Algebraic General Topology for definitions of used concepts.

Solved:

Coatoms of $ \mathsf{FCD}(A;B) $ are principal funcoids corresponding to binary relations of the form $ (A\times B)\setminus(\{x\}\times\{y\}) $ where $ x\in A $, $ y\in B $.

The set of funcoids is atomic but not atomistic.

See new version of the book Algebraic General Topology.

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