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 $L$.

Sources

• 1980: G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove and D.S. Scott: A Compendium of Continuous Lattices

• Mizar article WAYBEL10:def 7