Bialgebraic contexts for finite distributive lattices (Q1906530)

To every relation \(I \subseteq A \times B\) between two sets \(A\) and \(B\), a complete lattice can be associated, which is called the concept lattice of the formal context \((A,B,I)\). The elements of this lattice are certain pairs \((S,T)\) with \(S \subseteq A\) and \(T \subseteq B\); \(S\) then is the extent and \(T\) is the intent of \((S,T)\). The paper introduces the notion of a bialgebraic context \(((A;F), (B;G),I)\), where \((A;F)\) and \((B;G)\) are (partial) algebras such that the extents of \((A,B,I)\) are exactly the subalgebras of \((A;F)\) and the intents are just the subalgebras of \((B;G)\). A natural instance of this notion is \((V,V^*, \perp)\), where \(V\) is a finite-dimensional vector space, \(V^*\) is its dual, and \(\perp\) is the orthogonality relation. In the paper, a construction is described which yields for every finite distributive lattice \(\underline L : = (L; \vee, \wedge)\) a partial algebra \(({\mathcal D} (\underline L),*)\) with partial binary operation \(*\) and a relation \(I\) such that \(((L; \vee, \wedge), ({\mathcal D} (\underline L),*),I)\) is bialgebraic.