Lattices formed by unions of elements in an antichain Let $A_1, \dots, A_k$ be incomparable subsets (of $\{1, \dots, n\}$) and consider the poset $P$ consisting of all possible unions of these under inclusion. Its not hard to see that this is a lattice, and I am interested in the properties of this lattice.
(1) I am pretty sure that these lattices have been studied: does anyone have a reference to work in this area?
(2) Are these lattices geometric? Does the homology their order simplicial complex have a simple combinatorial description?
 A: There are a couple of characterizations of these lattices.
An element $x$ in a lattice $L$ is said to be an atom if $x>0$ and there is no $y$ with $0<y<x$. A subset $A$ of a complete lattice $L$ is said to be join dense $L=\{\bigvee R\mid R\subseteq A\}$. A complete lattice $L$ is said to be atomistic if the collection of atoms in $L$ is join dense.
A closure system on a set $X$ is a collection $C$ of subsets of $X$ that is closed under arbitrary intersection (including the empty intersection, so $X\in C$). An interior system on $X$ is a collection $I$ of subsets of $X$ that is closed under arbitrary union (and $\emptyset\in X$ since we consider the empty union).
The lattice $P$ in question is an atomistic lattice and every atomistic lattice is isomorphic to a lattice of the form $P$. Furthermore, if $X$ is a complete lattice, then $X$ is isomorphic to the interior system $\{(\uparrow x)^C\mid x\in X\}$. If $X$ is a complete lattice, and $A$ is the set of all atoms in $X$, then the set
$((\uparrow a)^C)_{a\in A}$ is an antichain, and if $X$ is atomistic, then every element in $\{(\uparrow x)^C\mid x\in X\}$ can be written as a union of elements in
$((\uparrow a)^C)_{a\in A}$.
If $C$ is a closure system on the set $X$, then define the dual closure operator $C^*:P(X)\rightarrow P(X)$ by letting $C^*(R)$ be the smallest element in $C$ with $R\subseteq C^*(R)$ whenever $R\subseteq X$ (i.e. $C^*(R)=\bigcap\{S\in C\mid R\subseteq S\}$).
If $(X,C)$ is a pair where $C$ is a closure system on the set $X$, then define a mapping $\iota:X\rightarrow C$ by letting $\iota(\{x\})=C^*(\{x\})$. Then $C$ is a complete lattice, and $\iota[X]$ is a join-dense subset of $X$. Therefore, let
$\Gamma(X,C)=(\iota,C)$.
If $\iota:A\rightarrow L$ is a function, $L$ is a complete lattice, and $\iota[A]$ is join-dense in $L$, then let $C=\{\iota^{-1}[\downarrow x]\mid x\in L\}$. Then $C$ is a closure system on the set $A$, so let $\Delta(\iota,L)=(A,C)$.
Then it is not too hard to show that $(X,C)=\Delta(\Gamma(X,C))$ whenever $C$ is a closure system on a set $X$ and if $L$ is a complete lattice and $\iota:A\rightarrow L$ with $\iota[A]$ join dense in $L$, then $(\iota,L)\simeq\Gamma(\Delta(\iota,L)).$ One can go further and show that $\Gamma,\Delta$ are actually equivalences of categories.
We say that a closure system $C$ on $X$ is $T_0$ if whenever $x,y\in X,x\neq y$, there is some $R\in C$ with $x\in R,y\not\in R$ or $x\not\in R,y\in R$. We observe that a closure system $C$ on $X$ is $T_0$ precisely when $\iota:X\rightarrow C$ is injective where $(\iota,C)=\Gamma(X,C)$. If $C$ is a closure system on $X$, then define the specialization ordering on $X$ by letting $x\leq y$ precisely when $C^*(\{x\})\subseteq C^*(\{y\})$. Then a closure system $C$ on $X$ is $T_0$ precisely when the specialization ordering is a partial ordering instead of simply a pre-ordering. We say that a closure system $C$ on $X$ is $T_1$ if $\{x\}\in C$ whenever $x\in X$. A closure system $C$ on $X$ is $T_1$ precisely when the specialization ordering on $X$ is simply the equality relation. We observe that $\Delta(L,A)$ is $T_1$ precisely when $A$ is the set of all atoms in $L$. This duality is an extension of the duality between geometric lattices and the definition of matroids in terms of closed sets.
Let $C=\{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,2,3\}\}$. Then $C$ is a $T_1$-closure system over the set $\{1,2,3\}$ which does not satisfy the exchange property, so $C$ is not a geometric lattice, but $C$ is atomistic.
A: As pointed out (and, perhaps, proved?) in Joseph's answer, the lattices you're considering are precisely the finite atomistic lattices. Indeed, your $A_i$ are the atoms of $P$ since they're pairwise incomparable. Conversely, in a finite atomistic lattice $L$ consider the subset $M$ of meet-irreducible elements: those covered by exactly one other element. Then the subsets $X_a=\{m\in M|m\not\ge a\}\subset M$ for all $a\in L$ form a lattice isomorphic to $L$ when ordered by inclusion. Moreover, each $X_a$ is a union of $X_b$ with $b$ an atom and such $X_b$ are pairwise incomparable. Atomistic lattices are well-studied and are considerably more general than geometric (i.e. finite atomistic upper-semimodular) lattices.
Update. I probably misunderstood the term "order simplicial complex". If you mean the simplicial complex formed by all chains in $P$, then the last question does seem highly nontrivial. One can find some work related to the topology of such complexes but I won't pretend to know whether this work answers your last question.
