📘 Closures and closed sets

In mathematics, a subset \(C\) of a set \(S\) is said to be closed under a closure operator \(\phi\) on \(S\) if applying the closure operator to \(C\) returns \(C\) itself: \(\phi(C) = C\).

In the GALACTIC framework closed sets are represented as instances of the Closed protocol that provide methods to check for membership and to retrieve the underlying elements of the closed set.

The galactic.algebras.closure.examples.color.core module defines the Colors closure operator (implementing the Closure protocol) that operates on colors represented by the Color elements.

from galactic.algebras.closure.examples.color.core import Colors
from galactic.algebras.lattice.examples.color.core import Color
colors = Colors([Color(red=1.0), Color(green=1.0)])
display(colors == Colors(colors))  # Closed set
display(list(colors))
display(Color(red=1.0) in colors)
display(Color(green=1.0) in colors)
display(Color(blue=1.0) in colors) # Not in the closed set
# Elements different from Color(red=1.0) and Color(green=1.0) are in the closed set
display(Color(red=1.0, green=1.0) in colors)
display(Color(red=0.5, green=0.2) in colors)
display(Color() in colors)

True

[Color(), Color(red=1, green=1)]

True

True

False

True

True

True