On implication bases in n-lattices

Abstract

Implication bases in n-lattices are not formally defined. We clarify the different types of implications we need to reconstruct a concept n-lattice and show they can be derived from the same set of implications. We use this to identify a particular type of implication base in n-contexts. Finally, we provide an algorithm for computing implicational closures with n-dimensional bases.

Towards Analyzing Variability in Space and Time of Products from a Product Line using Triadic Concept Analysis

“…As FCA, TCA computes concepts and implications. In TCA, a set of implications holds for each subset of dimensions [2,7], such as the implications between sets of snapshots' dates, implications between sets of features, and implications between pairs of dimensions e.g. (feature, snapshots' date).…”

Some Notes on Polyadic Concept Analysis

“…Different definitions of implications in triadic and polyadic contexts have been proposed through the years such as "Biedermann's implications" [11], attribute×condition, conditional attribute or attributional condition implications [12]. In [13], it was proposed to consider all the implications that hold in dyadic contexts resulting from combinations of two transformations of an n-context C (see Fig. 2): Note that possible implications include those in C ({greek,numbers},{latin}) whose support is in a Cartesian product of dimensions.…”

“…Such rules (under the more general umbrella of association rules) are already under consideration in the data mining community [14,15]. Additionally, if one is only interested in implications that do not contain the first dimension -the objects -these implications are all derivable from the implications of C ({S1},{S2,...,Sn}) through the application of Armstrong's axioms plus two other axioms, as discussed in [13]. This means that one only has to reason on a single type of implication.…”

“…Just as in the dyadic case, the set of all the implications that do not involve the first (object) dimension and that hold in an n-context (or any its implication bases) can be used to reconstruct the set of n-concepts restricted to their last n − 1 components [13].…”

“…3 (1, c)} is contextual in the first context only. In [13] is explained in an overly formal way, for which the author is very sorry, that constructing the features of all the n-concepts of an n-context only requires implications between "boxes", i.e. implications of the form…”