Whenever $P$ is a poset, write $L(P)$ for the collection of lowersets of $P$: $$L(P) := \{\lambda \subseteq P : \forall x,y \in P : x \leq y \,\&\, y \in \lambda \rightarrow x \in \lambda\}$$
With that notation in place, we can define "unicital complements" as follows, where $X$ is an arbitrary set:
$$L(\mathcal{P}(X)) \rightarrow L(\mathcal{P}(X)), \quad \lambda \mapsto \lambda^\perp$$ $$\lambda^\perp := \{x \in \mathcal{P}(X) : \forall y \in \lambda : |x \cap y| \leq 1\}.$$
In words: given a lowerset $\lambda$ of $\mathcal{P}(X)$, we get another lowerset $\lambda^\perp$ defined as the collection of all subsets of $X$ that intersect each element of $\lambda$ in at most one point. Call this the unicital complement of $\lambda$.
Mini Question. What is the "unicital complement" of a collection of subsets really called, and has it been studied anywhere? (Please answer in the comments section.)
Remark 0. This is an orthomap, or in other words an antitone Galois connection whose domain and codomain are identical.
Remark 1. If $P$ is a partially ordered set, then $\mathrm{Chain}(P)$ and $\mathrm{Antichain}(P)$ are unicital complements of each other.
Definition. Call $\lambda$ unicital-closed iff $(\lambda^\perp)^\perp = \lambda$.
Remark 2. Not every lowerset satisfies $(\lambda^\perp)^\perp = \lambda$. For example, let $X$ denote a set and let $\lambda$ denote the empty subset of $X$. Then $\lambda^\perp$ is the collection of all subsets of $X$, and hence $(\lambda^\perp)^\perp$ is the collection of all subsets of $X$ with at most one element.
Main Question. Given a set $X$, can we usefully characterize those lowersets $\lambda$ of $\mathcal{P}(X)$ that are unicital-closed?
Let $\lambda$ be in $L(\mathcal P(X)$. Define $\lambda_2$ to be the set of all the members of $\lambda$ that have exactly two elements. Note that $\lambda^\bot$ consists of exactly those subsets $x$ of $X$ that have no subset in $\lambda_2$.
As a consequence, if $\mu$ is another element of $L(\mathcal P(X))$, then $\lambda^\bot=\mu^\bot$ if and only if $\lambda_2=\mu_2$. Recall the general property of Galois connections that $\lambda^{\bot\bot}$ is the largest element $\mu$ in the poset such that $\lambda^\bot=\mu^\bot$. So in our situation, $\lambda^{\bot\bot}$ is the largest $\mu$ in $L(\mathcal P(X))$ such that $\lambda_2=\mu_2$.
That $\mu$ is simply the collection of those $x\subseteq X$ such that all $2$-element subsets of $x$ are in $\lambda_2$.
With this description of $\lambda^{\bot\bot}$, we get that $\lambda$ is unicital-closed iff, whenever it contains all $2$-element subsets of some $x\subseteq X$, then it also contains $x$.
Such families $\lambda$ have shown up in various contexts and, as a result, have various names. In connection with linear logic, Girard called them coherence spaces. They are also called clique complexes in graph theory. (The point is that $\lambda_2$ is the set of edges of a simple graph with vertex set $X$, and $\lambda=\lambda^{\bot\bot}$ consists all the cliques of that graph.) I've also seen such $\lambda$'s called flag complexes, but I don't remember where.
On the other hand, I don't recall having seen a name for the ${}^\bot$ operation.