I stumbled upon this, given some set $X$ and its powerset $\mathcal{P}(X)$ and some incomparable set $\mathbb{S}\subseteq\mathcal{P}(X)$, i.e. for any $S,S'\in\mathbb{S}$ we have $S\setminus S'\neq\emptyset$.
Then upward closure of $S$ is defined as $up(\mathbb{S})=\{S'\supseteq S\mid S\in\mathbb{S}\}$. Given an upward closed set its minimal elements define an incomparable set. Similarly for incomparable $\mathbb{T}\subseteq\mathcal{P}(X)$ we can define a downward closure $do(\mathbb{T})=\{T'\subseteq T\mid T\in\mathbb{T}\}$. And for any downward closed set its maximal elements define an incomparable set.
Now the interesting part starts: Each upward closed set $up(\mathbb{S})$ uniquely defines a downward closed set $do(\mathbb{S})=\mathcal{P}(X)\setminus up(\mathbb{S})$ and vice versa. Thus also each incomparable set of minimal elements uniquely defines an incomparable set of maximal elements and vice versa.
I was wondering if there is some research on this relations between such maximal and minimal sets. Cases where computation of one from the other allows for an efficient algorithm? Or knowledge on given cardinality of one do we know something about cardinality of the other?
The whole question seems so obvious but still I can not come up with an idea of in which area of mathematics to successfully look for results.
Just a few examples on how these maximal and minimal sets are related:
$\mathbb{S}_1=\{\{x\}\mid x\in X\}$, $\mathbb{T}_1=\emptyset$, cardinalities $\lvert\mathbb{S}_1\rvert=\lvert X\rvert$ and $\lvert\mathbb{T}_1\rvert=0$.
$\mathbb{S}_2=\{X\}$, $\mathbb{T}_2=\{\{X\setminus x\}\mid x\in X\}$, cardinalities $\lvert\mathbb{S}_2\rvert=1$ and $\lvert\mathbb{T}_2\rvert=\lvert X\rvert$.
$\mathbb{S}_3=\{\{x,y\}\mid x,y\in X,x\neq y\}$, $\mathbb{T}_3=\{\{x\}\mid x\in X\}$, cardinalities $\lvert\mathbb{S}_3\rvert={\lvert X\rvert\choose{2}}$ and $\lvert\mathbb{T}_3\rvert=\lvert X\rvert$.
$\mathbb{S}_4=\{\{a,b\},\{a,c\}\}$, $\mathbb{T}_4=\{\{a\},\{b,c\}\}$, cardinalities $\lvert\mathbb{S}_4\rvert=\lvert\mathbb{T}_4\rvert=2$.
$\mathbb{S}_5=\{\{a,b\},\{c,d\},\{e,f\}\}$, $\mathbb{T}_5=\{\{a,c,e\},\{a,c,f\},\{a,d,e\},\{a,d,f\},\{b,c,e\},\{b,c,f\},\{b,d,e\},\{b,d,f\}\}$, cardinalities $\lvert\mathbb{S}_5\rvert=3$ and $\lvert\mathbb{T}_5\rvert=8$.
generalization of (5): $\mathbb{S}_6=\{\{x_{1,i},x_{2,i}\}\mid 1\leq i\leq n\}$, $\mathbb{T}_6=\{\{x_{i,1},x_{j,2} ... x_{k,n}\}\mid i,j...k\in\{1,2\}\}$, cardinalities $\lvert\mathbb{S}_6\rvert=n$ and $\lvert\mathbb{T}_6\rvert=2^n$.
$\mathbb{S}_7=\{\{a,c,e\},\{a,c,f\},\{a,d,e\},\{a,d,f\},\{b,c,e\},\{b,c,f\},\{b,d,e\},\{b,d,f\}\}$, $\mathbb{T}_7=\{\{a,b,c,d\},\{a,b,e,f\},\{c,d,e,f\}\}$, cardinalities $\lvert\mathbb{S}_7\rvert=8$ and $\lvert\mathbb{T}_7\rvert=3$.
generalization of (7): $\mathbb{S}_8=\mathbb{T}_6$, $\mathbb{T}_8=\{\bigcup\mathbb{S}_8\setminus S\mid S\in\mathbb{S}_6\}$, cardinalities $\lvert\mathbb{S}_8\rvert=2^n$ and $\lvert\mathbb{T}_8\rvert=n$.
complementary to (4): $\mathbb{S}_9=\mathbb{T}_4=\{\{a\},\{b,c\}\}$, $\mathbb{T}_9=\{\{b\},\{c\}\}=\{X_9\setminus\{a,c\},X_9\setminus\{a,b\}\}$, cardinalities $\lvert\mathbb{S}_9\rvert=\lvert\mathbb{T}_9\rvert=2$.
not exactly complementary to (4): $\mathbb{S}_{10}=\{\{b,c\}\}$, $\mathbb{T}_{10}=\mathbb{S}_4=\{\{a,b\},\{a,c\}\}$, cardinalities $\lvert\mathbb{S}_{10}\rvert=1$ and $\lvert\mathbb{T}_{10}\rvert=2$.
Particulary Examples (7) and (8) highlight a few things:
In the general case there will not be polynomial time algorithms (where plain $\mathbb{S}$ or $\mathbb{T}$ serves as input and plain $\mathbb{T}$ or $\mathbb{S}$, resp. should be the output) for any direction. If the input is size $o(n)$ and the output is supposed to be $o(2^n)$ we need exponential time.
Looks like some kind complement transformation relation between incomparable sets where $\mathbb{S}$ is swapped with $\mathbb{T}$. Not sure though what the exact relation is. For instance $\mathbb{T}=\mathbb{S}_4$ does not work as a complentary version of Example (4), opposed to Example (9), illustrated in Example (10). I believe the complement construction however works if we consider complement minus $\bigcap\mathbb{T}$. Anyhow for this assumption we would need to fix or talk about $X$, $\bigcup\mathbb{S}$ and $\bigcap\mathbb{T}$ first.