Formal Concept Analysis (FCA) is an order-theoretic method for concept analysis and visualization, pioneered by Wille (Wille, 1982) in the mid-80s. A concept is a pair constituted by its extent and intent. The extent of a concept is a collection of all objects belonging to that concept and its intent is the set of all attributes common to all objects of the extent. FCA establishes the relationships between intents and extents and visualizes the generalization and specialization of concepts by means of concept lattices. Because of its strengths, FCA has been become a powerful tool for data mining (Mouakher and Yahia, 2019; Bartl and Konecny, 2019), social networks (Roth et al., 2008), software engineering (Tilley et al., 2005), cognition-based concept learning (Kuznetsov and Makhalova, 2018) and knowledge reduction (Konecny and Krajca, 2019; Cornejo et al., 2018).
In the study of knowledge representation and reasoning in FCA, (attribute) implication (Ganter and Wille, 1999; Qu and Zhai, 2008; Zhai et al., 2012; Ma et al., 2011; Zhai and Li, 2019) is proposed in the form of A→B, meaning that if all the attributes in A are satisfied, then all the attributes in B are satisfied. Duquenne etc. (V and J-L, 1986) constructed the so-called canonical basis, which turns out to be complete and non-redundant w.r.t. implication logic (Ganter and Wille, 1999; Stumme, 1996), and minimal among all complete sets of implications. Starting from a canonical basis of a formal context, one can obtain all implications in this context by applying Armstrong rules (Armstrong, 1974).