Abstract
The generalized one-sided concept lattices represent a generalization of the classical FCA method convenient for a hierarchical analysis of object-attribute models with different types of attributes. The mentioned types of object-attribute models are formalized within the theory as formal contexts of a certain type. The aim of this paper is to investigate some intercontextual relationships represented by the notion of bond. A composition of bonds is defined in order to introduce the category of formal contexts with bonds as morphisms. It is shown that there is a one-to-one correspondence between bonds and supremum preserving mappings between the corresponding generalized one-sided concept lattices. As the main theoretical result it is shown that the introduced category of formal contexts with bonds is equivalent to the category of complete lattices with supremum preserving mappings as morphisms.
Highlights
The theory of concept lattices or formal concept analysis (FCA for short) represents a method of data analysis for identifying conceptual structures among data sets
The notion of bond within the theory of generalized one-sided concept lattices was introduced. For these types of concept lattices, a bond between two formal contexts is a formal context with the object set of the former one and with the attribute set from the second one
The main construction in various FCA-based methods of hierarchical analysis is a creation of the concept lattice corresponding to some object-attribute model
Summary
The theory of concept lattices or formal concept analysis (FCA for short) represents a method of data analysis for identifying conceptual structures among data sets. The name “one-sided” refers to the fact that the output concepts are formed by crisp subsets of objects and vectors of fuzzy values, characterizing the objects in concepts This type of concept lattices was studied by several authors, e.g., in [7] an extension concerning preference relation on attributes was described, papers [8,9] deal with problems of attribute reductions, while [10,11] deal with alternative definitions of concept forming operators. As in the classical case, such definition of the notion of a bond is possible, since objects and attributes are evaluated in a single structure (residuated lattice L) and concepts consist of pairs of L-sets. Conclusion section, some potential theoretical application of the introduced notion is discussed, e.g., in a reduction process of the mentioned types of concept lattices
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