Formal query systems on contexts and a representation of algebraic lattices

Formal Concept Analysis (FCA) has proven to be a feasible framework for information retrieval (IR). In this paper, with the aim of exploring the mathematical properties of the FCA-based document retrieval paradigm, we propose the notions of conjunctive and disjunctive query on the formal contexts and investigate the derived query systems from the order-theoretical perspective. We show that the conjunctive query system is isomorphic to the Galois lattice of the underlying context, while the disjunctive query system can be join-densely generated from the Galois lattice of the underlying context up to isomorphism. Next, we introduce directed disjunctive query systems and study their connection with algebraic lattices. As a result, we show that the notion of directed disjunctive query provides an alternative approach to restructuring algebraic lattices.