Operators#

GALACTIC core closure

A closure operator is a function that maps an element of a partially ordered set (poset) to another element in the same set, ensuring that the result is closed in a specific sense.

Set theoretic operators#

Definition

For a set \(X\), we denote \(2^X\) as the set of subsets of \(X\). We recall that a set-theoretic operator \(\phi\) on \(X\) is a function from \(2^X\) to \(2^X\).

Properties

Extensibility

The operator \(\phi\) is said to be extensive if:

\[ \forall A \subseteq X, A \subseteq \phi(A) \]

Contractivity

The operator \(\phi\) is said to be contractive if:

\[ \forall A \subseteq X, A \supseteq \phi(A) \]

Isotonicity

The operator \(\phi\) is said to be isotone (or monotone) if:

\[ \forall A, B \subseteq X, A\subseteq B \implies \phi(A) \subseteq \phi(B) \]

Antitonicity

The operator \(\phi\) is said to be antitone if:

\[ \forall A, B \subseteq X, A\subseteq B \implies \phi(A) \supseteq \phi(B) \]

Idempotence

The operator \(\phi\) is said to be idempotent if:

\[ \forall A \subseteq X, \phi\circ\phi(A) = \phi(A) \]

Normalization

The operator \(\phi\) is said to be normalized if:

\[ \phi(\emptyset)=\emptyset \]

Anti-exchange

The operator \(\phi\) is said to have the anti-exchange property if:

\[ \forall A\subseteq X, \forall p \neq q \in X \setminus \phi(A), p \in \phi(\{q\}\cup A) \implies q \notin \phi(\{p\} \cup A) \]

Atomicity

The operator \(\phi\) is said to be atomic if it is normalized and:

\[ \forall p\in X, \phi(\{p\}) = \{p\} \]

Algebraic

The operator \(\phi\) is said to be algebraic if:

\[ \forall A, B\subseteq X, \phi(A)=\bigcup\{\phi(B)\;|\;B\subseteq A \mbox{ finite}\} \]

Closure operators#

Definition

Let \(X\) be a ground set. An operator \(\phi\) is called a closure operator if it satisfies the following properties:

Extensitivity: The operator is extensive.
Idempotence: The operator is idempotent.
Isotonicity: The operator is isotone.

Let \(A\subset X\), \(\phi(A)\) is called a closed set.

Moore Families#

The set of images by \(\phi\) of all subsets of \(X\) forms a closure system or Moore family. It is a collection of subsets that is closed under arbitrary intersections.

Antitone Galois connection#

An (antitone) Galois connection between the parts of \(X\) and the parts of \(Y\) is a couple of antitone functions \(\left\langle\alpha: 2^X\to 2^Y, \beta: 2^Y\to 2^X \right\rangle\) such that:

\[ \forall A \subseteq X,B\subseteq Y, B \subseteq \alpha(A) \iff A \subseteq \beta(B) \]

The compositions of \(\alpha\) and \(\beta\) form closure operators on \(X\) and \(Y\) (\(\beta\circ\alpha\) is a closure operator on \(X\) and \(\alpha\circ\beta=\) is a closure operator on \(Y\)).