# Definition:Ordered Set of Subalgebras

## Definition

Let $L$ be an ordered set.

The ordered set of subalgebras of $L$ is ordered subset of $\operatorname{ClSystems}\left({L}\right)$ and is defined by

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

where

$X$ is the set of all directed suprema inheriting closure systems on $L$,
$\operatorname{ClSystems}\left({L}\right)$ denotes the ordered set of closure systems on $L$.