📘 Moore families

In mathematics, a Moore family (or a closure system) on a set \(E\) is a family \(\mathcal{F}\) of subsets of \(E\) that:

1. Contains \(E\) itself: \(E \in \mathcal{F}\)

2. Is closed under arbitrary intersections: for any subfamily

\(\mathcal{G} \subseteq \mathcal{F}\), we have \(\bigcap \mathcal{G} \in \mathcal{F}\)

The intersection property ensures that for any subset \(A \subseteq E\), there exists a smallest element of \(\mathcal{F}\) containing \(A\), called the closure of \(A\).

Examples

Using the ExtensibleMooreFamily (or the frozen FrozenMooreFamily) from the galactic.algebras.closure.core module, you can create Moore families:

from IPython.display import HTML
from galactic.algebras.closure.core import ExtensibleMooreFamily
from galactic.algebras.lattice.examples.color.core import Color
from galactic.algebras.closure.examples.color.core import Colors
family = ExtensibleMooreFamily()

family.extend(
    [
        Colors([Color(red=1, blue=0.5), Color(blue=1, red=0.5)]),
        Colors([Color(blue=1, green=0.5), Color(green=1)]),
    ],
)
display(*family.essentials)

Moore families are lattices of closed sets and can be visualized in the GALACTIC framework:

from galactic.algebras.closure.examples.color.viewer import ColorsRenderer
from galactic.algebras.poset.viewer import HasseDiagram
diagram = HasseDiagram[Colors](family, domain_renderer=ColorsRenderer())
diagram

In the GALACTIC framework, you can get the join-irreducible and meet-irreducible closed sets of a Moore family using the join_irreducible and meet_irreducible properties:

display(*family.join_irreducible, HTML("<br>"))
display(*family.meet_irreducible)

It’s possible to get the principal closed set where an element appears for the first time using the principals property:

for elem, closed in family.principals.items():
    display(elem, closed, HTML("<br>"))

family.extend(
    [
        Colors([Color(red=1), Color(green=0.5)]),
    ]
)
for elem, closed in family.principals.items():
    display(elem, closed, HTML("<br>"))
diagram

The irreducibles are:

display(*family.join_irreducible, HTML("<br>"))
display(*family.meet_irreducible)

It’s also possible to get the elements introduced by a closed set in the family using the introduced_elems() method:

for closed in family:
    display(closed, *family.introduced_elems(closed), HTML("<br>"))