Paper:

# Fuzzy Concept Lattices Constrained by Hedges

## Radim Belohlavek^{*,**} and Vilem Vychodil^{**}

^{*}Dept. Systems Science and Industrial Engineering, T. J. Watson School of Engineering and Applied Science Binghamton University - SUNY, Binghamton, NY 13902-6000, USA

^{**}Dept. Computer Science, Palacky University, Olomouc, Tomkova 40, CZ-779 00 Olomouc, Czech Republic

We study concept lattices constrained by hedges. The principal aim is to control, in a parameterical way, the size of concept lattices, i.e. the number of conceptual clusters extracted from data. The paper presents theoretical insight, comments, and examples. We introduce new, parameterized, concept-forming operators and study their properties. We obtain an axiomatic characterization of the concept-forming operators. Then, we show that a concept lattice with hedges is indeed a complete lattice which is isomorphic to an ordinary concept lattice. We describe the isomorphism and its inverse. These mappings serve as translation procedures. As a consequence, we obtain a theorem characterizing the structure of concept lattices with hedges which generalizes the well-known main theorem of ordinary concept lattices. Furthermore, the isomorphism and its inverse enable us to compute a concept lattice with hedges using algorithms for ordinary concept lattices. Further insight is provided for boundary choices of hedges. We demonstrate by experiments that the size reduction using hedges as parameters is smooth.

*J. Adv. Comput. Intell. Intell. Inform.*, Vol.11, No.6, pp. 536-545, 2007.

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