Antitone Galois connection

Polarities

In mathematics an antitone Galois connection is a couple of antitonic functions (called polarities) \((\alpha, \beta)\) from the power set of one set to the power set of another set: \(\alpha:2^E\rightarrow 2^F\), \(\beta:2^F\rightarrow 2^E\)

such that:

\(f \subseteq \alpha(e) \subseteq F\) if and only if \(e \subseteq \beta(f) \subseteq E\)

from galactic.algebras.examples.connection import IntegerConnection

connection = IntegerConnection([2, 3, 4, 5, 6], [12, 18, 24, 36, 48, 30, 60])

connection.polarities

(<galactic.algebras.examples.connection.MultiplePolarity at 0x7f3bdc188610>,
 <galactic.algebras.examples.connection.DivisorPolarity at 0x7f3bdc187e00>)

list(connection.polarities[0](elements={2, 5}))

[30, 60]

list(connection.polarities[1](elements={30}))

[2, 3, 5, 6]

Closures

If \((\alpha, \beta)\) is an antitone Galois connection from \(2^E\) to \(E^S\), \(\beta \circ \alpha\) and \(\alpha \circ \beta\) are closure operators on respectively \(E\) and \(F\).

connection.closures

(compose(<galactic.algebras.examples.connection.DivisorPolarity object at 0x7f3bdc187e00>, <galactic.algebras.examples.connection.MultiplePolarity object at 0x7f3bdc188610>),
 compose(<galactic.algebras.examples.connection.MultiplePolarity object at 0x7f3bdc188610>, <galactic.algebras.examples.connection.DivisorPolarity object at 0x7f3bdc187e00>))

list(connection.closures[0](elements={2, 3}))

[2, 3, 6]

list(connection.closures[1](elements={30}))

[30, 60]