Ordered Set of Closure Operators and Dual Ordered Set of Closure Systems are Isomorphic

Theorem

Let $L = \left({S, \vee, \wedge, \preceq}\right)$ be a complete lattice.

Then $\operatorname{Closure}\left({L}\right)$ and $\operatorname{ClSystems}\left({L}\right)^{-1}$ are order isomorphic

where

$\operatorname{Closure}\left({L}\right)$ denotes the ordered set of closure operators of $L$,

$\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems oj $L$,

$\operatorname{ClSystems}\left({L}\right)^{-1}$ denotes the dual to $\operatorname{ClSystems}\left({L}\right)$.

Proof

By definition of ordered set of closure operators:

$\operatorname{Closure}\left({L}\right) = \left({X, \preceq'}\right)$

where

$X$ is the set of all closure operators on $L$,

for all closure operators $f, g$ on $L$: $f \preceq' g \iff f \preceq g$

$\preceq$ denotes the ordering on mappings.

By definition of ordered set of closure systems:

$\operatorname{ClSystems}\left({L}\right) = \left({Y, \precsim}\right)$

where

$Y$ is the set of all closure systems on $L$,

for all closure systems $s_1 = \left({T_1, \preceq_1}\right)$, $s_2 = \left({T_2, \preceq_2}\right)$ on $L$: $s_1 \precsim s_2 \iff T_1 \subseteq T_2$

Define $F:X \to Y$ by Image of Closure Operator Inherits Infima:

for all closure operators $f$ on $L$: $F\left({f}\right) = \left({f\left[{S}\right], \mathord\preceq \cap \left({f\left[{S}\right] \times f\left[{S}\right]}\right)}\right)$

By Operator Generated by Closure System is Closure Operator and Image of Operator Generated by Closure System is Set of Closure System:

$F$ is a surjection.

By Ordering on Closure Operators iff Images are Including:

$F$ is an order embedding.

Thue by definition:

$F$ is an order isomorphism.

$\blacksquare$