m-Algebraic lattices in formal concept analysis

The notion of an m -algebraic lattice, where m stands for a cardinal number, includes numerous special cases, such as complete lattice, algebraic lattice, and prime algebraic lattice. In formal concept analysis, one fundamental result states that every concept lattice is complete, and conversely, each complete lattice is isomorphic to a concept lattice. In this paper, we introduce the notion of an m -approximable concept on each context. The m -approximable concept lattice derived from the notion is an m -algebraic lattice, and conversely, every m -algebraic lattice is isomorphic to an m -approximable concept lattice of some context. Morphisms on m -algebraic lattices and those on contexts are provided, called m -continuous functions and m -approximable morphisms, respectively. We establish a categorical equivalence between LAT m , the category of m -algebraic lattices and m -continuous functions, and CXT m , the category of contexts and mapproximable morphisms.We prove that LAT m is cartesian closed whenevermis regular and m > 2. By the equivalence of LAT m and CXT m , we obtain that CXT m is also cartesian closed under same circumstances. The notions of a concept, an approximable concept, and a weak approximable concept are showed to be special cases of that of an m -approximable concept.