Complement Operation between Union and Intersection Power Structures is Isomorphism

Theorem
Let $S$ be a set and let $\powerset S$ be its power set.

Let $\struct {\powerset S, \cap, \subseteq}$ be the ordered semigroup formed from the set intersection operation and subset relation.

Let $\struct {\powerset S, \cap, \supseteq}$ be the ordered semigroup formed from the set intersection operation and superset relation.

Let $\struct {\powerset S, \cup, \subseteq}$ be the ordered semigroup formed from the set union operation and subset relation.

Let $\struct {\powerset S, \cup, \supseteq}$ be the ordered semigroup formed from the set union operation and superset relation.

Let $\complement: \powerset S \to \powerset S$ be the complement operation on $\powerset S$:
 * $\forall X \in \powerset S: \map \complement X = S \setminus X$

where $S \setminus X$ denotes the set difference of $X$ with $S$.

Then $\complement$ is an ordered semigroup isomorphism from:
 * $\struct {\powerset S, \cap, \subseteq}$ to $\struct {\powerset S, \cup, \supseteq}$

and:
 * $\struct {\powerset S, \cap, \supseteq}$ to $\struct {\powerset S, \cup, \subseteq}$

Proof
From:
 * Power Set with Intersection and Subset Relation is Ordered Semigroup
 * Power Set with Intersection and Superset Relation is Ordered Semigroup
 * Power Set with Union and Subset Relation is Ordered Semigroup
 * Power Set with Union and Superset Relation is Ordered Semigroup

each of the given ordered structures is indeed an ordered semigroup.

From Relative Complement Mapping on Powerset is Bijection, $\complement$ is a bijection.

From Relative Complement of Relative Complement it follows that $\complement$ is an involution.

From De Morgan's Laws for Relative Complements:
 * $\map \complement {T_1 \cap T_2} = \map \complement {T_1} \cup \map \complement {T_2}$

and:
 * $\map \complement {T_1 \cup T_2} = \map \complement {T_1} \cap \map \complement {T_2}$

hence demonstrating that $\complement$ exhibits the morphism property from both $\struct {\powerset S, \cap}$ to $\struct {\powerset S, \cup}$ and back again.

Then from Relative Complement inverts Subsets we have that $\complement$ is an order isomorphism between $\struct {\powerset S, \subseteq}$ and $\struct {\powerset S, \supseteq}$ in both directions.

Hence the result.