Central points and approximation in residuated lattices

Radim Belohlavek, Michal Krupka.

Given a subset B of a complete residuated lattice, what are its points which are reasonably close to any point of B? What are the best such points? In this paper, we seek to answer these questions provided closeness is assessed by means of biresiduum, i.e. the truth function of equivalence in fuzzy logic. In addition, we present two algorithms which output, for a given input set M of points in a residuated lattice, another set K which approximates M to a desired degree. We prove that the algorithms are optimal in that the set K is minimal in terms of the number of its elements. Moreover, we show that the elements of any set K' with such property are bounded from below and from above by the elements produced by the two algorithms.

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