Closure Operator Preserves Directed Suprema iff Image of Closure Operator Inherits Directed Suprema

Theorem

Let $L = \struct {S, \vee, \wedge, \preceq}$ be a complete lattice.

Let $h: S \to S$ be a closure operator on $L$.

Then $h$ preserves directed suprema if and only if $\struct {h \sqbrk S, \precsim}$ inherits directed suprema.

where

$h \sqbrk S$ denotes the image of $h$,

$\mathord\precsim = \mathord\preceq \cap \paren {h \sqbrk S \times h \sqbrk S}$

Proof

By Operator Generated by Image of Closure Operator is Closure Operator:

$\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} } = h$

where $\map {\operatorname{operator} } {\struct {h \sqbrk S, \precsim} }$ denotes the operator generated by $\struct {h \sqbrk S, \precsim}$

Hence the result holds by Operator Generated by Closure System Preserves Directed Suprema iff Closure System Inherits Directed Suprema.

$\blacksquare$

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 WAYBEL10:25