General Concept Lattice

[1][2][3] The formal context is a data table of heterogeneous relations illustrating how objects carrying attributes.

By analogy with truth-value table, every formal context can develop its fully extended version including all the columns corresponding to attributes constructed, by means of Boolean operations, out of the given attribute set.

[4] The GCL[4] claims to take into account the extended formal context for preservation of information content.

In contrast, the GCL is an invariant lattice structure with respect to these formal contexts since they can infer each other and ultimately entail the same information content.

In information science, the Formal Concept Analysis (FCA) promises practical applications in various fields based on the following fundamental characteristics.

Alternative concept lattices subject to different derivation operators based on the notions relevant to the Rough Set Analysis have also been proposed.

The GCL accomplishes a sound theoretical foundation for the concept hierarchies acquired from formal context.

[4] Maintaining the generality that preserves the information, the GCL underlies both the FCL and RSL, which correspond to substructures at particular restrictions.

Technically, the GCL would be reduced to the FCL and RSL when restricted to conjunctions and disjunctions of elements in the referred attribute set (

In addition, the GCL unveils extra information complementary to the results via the FCL and RSL.

Surprisingly, the implementation of formal context via GCL is much more manageable than those via FCL and RSL.

The derivation operators constitute the building blocks of concept lattices and thus deserve distinctive notations.

, are considered as different modal operators[7][8] (Sufficiency, Necessity and Possibility, respectively) that generalise the FCA.

, the operator adopted in the standard FCA,[1][2][3] follows Bernhard Ganter [de][10] and R. Wille;[1]

Note that (1) and (2) enable different object-oriented constructions for the concept hierarchies FCL and RSL, respectively.

Note that (3) corresponds to the attribute-oriented construction[9] where the roles of object and attribute in the RSL are exchanged.

based on the RSL in a similar manner:[4] the set of all objects carrying any of the attributes in

For every formal context one may acquire its extended version deduced in the sense of completing a truth-value table.

The FCL and RSL will not be altered if their intents are interpreted as single attributes.

which exhausts all the possible unions of the minimal object sets discernible by the formal context.

Note that the GCL also appears to be a Hasse diagram due to the resemblance of its extents to a power set.

also exhibits another Hasse diagram isomorphic to the ordering of attributes in the closed interval

from the formal context was known to be complicated,[13][14][15][16][17] it necessitates efforts for constructing a canonical basis, which does not apply to the implications of type

By contrast, the above equivalence only proposes[4] Hence, purely algebraic formulae can be employed to determine the implication relations, one need not consult the object-attribute dependence in the formal context, which is the typical effort in finding the canonical basis.

Notably, the point of view conjunction-to-conjunction has also been emphasised by Ganter[5] while dealing with the attribute exploration.

One could overlook significant parts of the logic content in formal context were it not for the consideration based on the GCL.

Here, the formal context describing 3BS given in Table 1 suggests an extreme case where no implication of the type

Notably, some of these rules can be reduced to the Armstrong axioms, which pertain to the main considerations of Guigues and Duquenne[6] based on the non-redundant collection of informative implications acquired via FCL.

Note that one also arrives at For concreteness, consider the example depicted by Table 2, which has been originally adopted for clarification of the RSL[9] but worked out for the GCL.

Moreover, informative implications could also relate different nodes via Hypothetical syllogism by invoking tautology.

Table 1
Fig. 1 : Three different formal concept lattices (FCLs) obtained from the three formal contexts describing the same 3BS , where balls are equipped with three distinct colours.
Fig. 2 : Three different rough set lattices, cf. Fig. 1, obtained from the same three formal contexts describing the 3BS .
Fig. 3: Identifying FCL and RSL on the GCL for the 3BS according to the formal context in Table 1 . Every general intent comprises all the attributes uniquely possessed by the object set in common. Elements on can be ordered as a Hasse diagram identifiable with the closed interval where .
Fig. 4: Readout of GCL from a formal context. On each node, a binary string is to denote the extent, e.g., 01010 denotes the object set i.e. , 00100 denotes i.e. . In this figure, and are shown. Accordingly, one may identify all the lower and upper bounds of intents in the expressions the contextual truth and falsity ( and ), respectively.
The GCL constructed according to a formal context.
Fig. 5: The GCL constructed according to the formal context given in Table 2 . The circled points are nodes existing on the FCL, whereas the bold ones belong to the RSL, also cf. Fig. 3 .