Formal concept analysis (FCA) has a significant appeal as a formal framework for knowledge discovery not least because of the mathematical tools it provides for a range of data manipulations such as splits and merges. We study the computation of the canonical basis of a context starting from the bases of two apposed subcontexts, called factors. Improving on a previous method of ours, we provide here a deeper insight into its pivotal implication family and show it represents a relative basis. Further structural results allow for more efficient computation of the global basis, in particular, the relative one admits, once added to factor bases, an inexpensive reduction. A method implementing the approach as well as a set of further combinatorial optimizations is shown to outperform NextClosure on at least one dataset.
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