Formal Concept Analysis

Dualization in lattices given by ordered sets of irreducibles

“…An element X ∈ L is called infimum-irreducible (or meet-irreducible) if X = Y >X Y , i.e., X is not represented by the intersection of any elements above it. Dually, an element X ∈ L is called supremum- In what follows we use the standard definitions and facts of Formal Concept Analysis (FCA) from [11]. Let G and M be sets, called the set of objects and attributes, respectively.…”

“…A well-known fact is that every lattice is determined up to isomorphism by the ordered set of its meet (infimum) and join (supremum) irreducible elements [11]. These elements cannot be represented as meets (joins) of other elements that are larger (smaller) then them.…”

“…On diagram of finite lattices these elements have one upper (lower) neighbor. In this paper we consider finite lattices given by ordered sets of their meet and join irreducibles, known as concept lattices [11,1,14]. We show that dualization for representation of this type is impossible in output polynomial time unless P = NP.…”

“…In what follows we shall use the notation of Formal Concept Analysis [11], which provides a convenient language and necessary results for lattices given by ordered sets of irreducible elements.…”

Mining Association Rules in Folksonomies

“…I. e., ≺ = ∅, and we will simply note a folksonomy as a quadruple F := (U, T, R, Y ). This structure is known in Formal Concept Analysis (Wille (1982); Ganter and Wille (1999)) as a triadic context (Lehmann and Wille (1995);Stumme (2005)). An equivalent view on folksonomy data is that of a tripartite (undirected) hypergraph G = (V, E), where V = U∪T∪R is the set of nodes, and E = {{u, t, r} | (u, t, r) ∈ Y } is the set of hyperedges.…”