Further Galois Connections between Semimodules over Idempotent Semirings

In [14] a generalisation of Formal Concept Analysis was introduced with data mining applications in mind, K-Formal Concept Analysis, where incidences take values in certain kinds of semirings, instead of the standard Boolean carrier set. The construction leading to the pair of dually (order) isomorphic lattices can be further manipulated to obtain the three other types of Galois Connections providing a fuller set of tools to interpret any relations between data. We relate this result to previous descriptions of certain instances of such Galois Connections in qualitative data analysis and provide concrete examples of them related to Rmax,+-semimodules in quantitative data analysis. 1 Motivation: Lattices related to an Incidence Data analysis results improve when many different tools are offered to the practitioner. Consider then the modal operators ([13], def. 3.8.2; [6]) introduced by a Boolean matrix, I ∈ 2G×M , over a set of objects, A ∈ 2G and, dually, over sets B ∈ 2M of attributes operated by the converse relation It ∈ 2M×G as listed in Table 1. Formal Concept Analysis adepts may recognise the extent and intent polars in the sufficiency operators for a relation, [[I ]] (A) = A′, [[It]] (B) = B′ , but also their closure operators, [[It]][[I ]](A) = A′′ , [[I ]] [[It]] (B) = B′′ . Perhaps less known is that the pairs of operators in the first and second rows of Table 1 define the neighbourhood lattices: For a formal context (G, M, I) define the span of a set of objects as: span(A) := 〈I〉(A) = (A)∃ . This is the set of attributes related to some g ∈ A 1. Similarly, define for its dual context (M,G, It) the content of a set of attributes, content(B) = [It] (B) = (B)I , as the set of objects which can be completely described by the attributes in B . Next consider the set N(G, M, I) (for Ger. Nachbar, neighbour) of neighbour pairs, (A,B) ∈ N(G, M, I), such that span(A) = (A)∃ = B ⇔ A = (B)I = content(B) . Then we can state the: ! This work has been partially supported by a grants from the Spanish GovernmentComisión Interministerial de Ciencia y Tecnoloǵıa project TEC2005-04264/TCM. 1 The second, operator notation is closer to Galois connection theory as explained below and relates better to normal notation in Formal Concept Analysis. Table 1. Modal operators over a relation and its converse for sets of objects A ⊆ G and attributes B ⊆ M . The misalignment in the first two rows is intentional. possibility operator over G necessity operator over M 〈I〉 (A) = {m ∈ M | (∃g ∈ G)[g ∈ A ∧ gIm] } ˆ It ̃ (B) = { g ∈ G | (∀m ∈ M)[mItg ⇒ m ∈ B) } necessity operator over G possibility operator over M [I] (A) = {m ∈ M | (∀g ∈ G)[gIm ⇒ g ∈ A) } ̇ It ̧ (B) = { g ∈ G | (∃m ∈ M)[m ∈ B ∧ mItg] } sufficiency operator over G sufficiency operator over M [[I]] (A) = {m ∈ M | (∀g ∈ G)[g ∈ A ⇒ gIm) } ˆˆ It ̃ ̃ (B) = { g ∈ G | (∀m ∈ M)[m ∈ B ⇒ mItg) } dual sufficiency operator over G dual sufficiency operator over M 〈〈I〉〉 (A) = {m ∈ M | (∃g ∈ G)[g / ∈ A ∧ g"I"m] } ̇ ̇ It ̧ ̧ (B) = { g ∈ G | (∃m ∈ M)[m / ∈ B ∧ m "I"t g] } Theorem 1 (Fundamental theorem of Neighbourhood lattices [6]). The neighbourhood lattice, N(G, M, I), is a complete lattice in which infimum and supremum are given by: