multiplex (version 4.0)

galois: Galois Derivations

Description

Function to compute Galois derivations between partially ordered subsets.

Usage

galois(x, labeling = c("full", "reduced"), sep)

Value

A list of “Galois” class object with Galois derivations of objects and attributes as list labels:

sep

pair separator

gc

galois connections of pairs

Notice that the object attribute class only is printed for "full" labeling, which is an intrinsic part of the outcome for a "reduced" labeling.

Arguments

x

An incidence matrix with objects and attributes.

labeling

Type of labeling for the Galois derivations:

• full

• reduced

sep

(optional) Pair separator for the pairwise relations, default “comma-space”.

Details

A Galois derivation, or Galois connection, between the power sets of \(G\) and \(M\) is defined for subsets \(A \subseteq G\) and \(B \subseteq M\) by the set of attributes common to all objects in \(A\), or \(A^\prime\), and the set of objects having all attributes in \(B\), or \(B^\prime\).

An inclusion with the "full" labeling in the output implies first objects and then attributes in the Galois connection, whereas with the "reduced" option the output assumes inclusions of attributes first and then the objects.

Notice that for many-valued formal contexts the structure is derived from a concept scaling through function cscl where is possible to define a pair separator between attributes and their values. A warning message is produced when there is a conflict between such separators and the pair separator chosen in sep for extents and intents.

References

Ganter, B. and R. Wille Formal Concept Analysis -- Mathematical Foundations. Springer. 1996.

See Also

partial.order, cscl, diagram, fltr.

Examples

# incidence matrix as a data frame
dfr <- data.frame(x=1:3, y=5:7)

# compute galois derivations 
galois(dfr)