# Definition:Ordered Set of Closure Systems

## Definition

Let $L$ be an ordered set.

The ordered set of closure systems of $L$ is a relational structure

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

where

$X$ is the set of all closure systems of $L$,
dor all closure systems $S_1 = \left({T_1, \preceq_1}\right), S_2 = \left({T_2, \preceq_2}\right)$ of $L$: $S_1 \precsim S_2 \iff T_1 \subseteq T_2$