Semiconcept and concept representations

In FCA, we often deal with a formal context $K=(G,M,I)$ that is only partially known, i.e. only the attributes that belong to an observable set $N\subset M$ are known. There must also exist a part $H$ of the object set $G$ – called a training set – that consists of elements with all attributes known. The concepts of $K$ have to be determined using the subcontexts corresponding to the training object set $H$ and to the observable attribute set $N$. In our paper, this problem is examined within the extended framework of the semiconcepts of the original context, which are generalizations of its concepts. Each semiconcept of the original context induces a semiconcept in both subcontexts. In this way, each semiconcept of the context is represented by an induced pair of semiconcepts, which can also be considered its approximations — as in the case of rough sets. We describe the properties of the mapping defined by this representation and prove that the poset formed by these semiconcept pairs is a union of two complete lattices. We show that these induced semiconcept pairs can be generated by using a simplified representation of them. As the number of semiconcepts grows exponentially with the size of the training set and the observable attribute set, an algorithm that selects the representation pairs for which their support and relevance reach a certain threshold is also presented.