Editing Formal concept analysis (section)

== Implications ==
An ''[[Implication (information science)|implication]]'' ''A'' → ''B'' relates two sets ''A'' and ''B'' of attributes and expresses that every object possessing each attribute from ''A'' also has each attribute from ''B''. When {{math|(''G'',''M'',''I'')}} is a formal context and ''A'', ''B'' are subsets of the set ''M'' of attributes (i.e., ''A,B'' ⊆ ''M''), then the implication ''A'' → ''B'' ''is valid'' if ''{{prime|A}}'' ⊆ ''{{prime|B}}''. For each finite formal context, the set of all valid implications has a ''canonical basis'',<ref>{{cite journal |last1=Guigues |first1=J.L. |last2=Duquenne |first2=V. |title=Familles minimales d'implications informatives résultant d'un tableau de données binaires |journal=Mathématiques et Sciences Humaines |volume=95 |issue= |pages=5–18 |date=1986 |doi= |url=http://www.numdam.org/item/MSH_1986__95__5_0.pdf}}</ref> an irredundant set of implications from which all valid implications can be derived by the natural inference ([[Armstrong axioms|Armstrong rules]]). This is used in ''attribute exploration'', a knowledge acquisition method based on implications.<ref name="GanterObiedkov">{{cite book |last1=Ganter |first1=Bernhard |last2=Obiedkov |first2=Sergei |title=Conceptual Exploration |publisher=Springer |date=2016 |isbn=978-3-662-49290-1 |pages= |url=}}</ref>