A *formal context* in *formal concept analysis* is a triple $K = (G, M, I)$ where $G$ is a set of objects, $M$ is a set of attributes and the binary relation $I \subset G \times M$ shows which objects possess which attributes. By single sorted case, I refer to the situation that $G=M$. One example of this situation is the fact that for a poset $P$, the *concept lattice* $\mathcal B(P,P,R_\leq)$ yields the Dedekind-MacNeille completion of $P$.

The concrete construction which I want to better understand is closely related to this example: For a given partial order $\leq$, define the relation $\tilde <$ via $x\tilde<y$ if and only if $x\leq y$ and $y\not \leq x$ or $x$ is a bottom element or $y$ is a top element. I want to understand the relation between $\mathcal B(P,P,R_\leq)$ and $\mathcal B(P,P,R_{\tilde<})$, as well as what happens to the relation between $P$ and $\mathcal B(P,P,R_{\tilde<})$ when I iterate this construction.

I noticed that $R_{\tilde<}$ is still a transitive relation, and that the binary relation $I$ will probably be transitive for all instances of the single sorted case of interest to me. Hence my question is whether the single sorted case of formal concept analysis been investigated already (especially with a transitive incidence relation).

**Additional remark** The single sorted case is a special case of the general theory, but the general theory should be embeddable in the single sorted case via $K'=(V, V, \imath_{V\times V}(I))$ with $V:=G \uplus M$ where $\imath_{V\times V}(I)$ interprets the relation $I \subset G \times M$ as a relation on $V\times V$. The incidence relation $\imath_{V\times V}(I)$ is transitive, but fails to be a partial order. On second thought, the embedding would also work with $V:=G \cup M$, but the incidence relation $\imath_{V\times V}(I)$ can fail to be transitive in this case.