Abstract
In Section 3.6 of Fuzzy relation equations and their applications to knowledge engineering. Kluwer Academic Publishers, Boston (1989), Di Nola et al. presented a procedure to find a minimal solution from a fixed solution of a system of fuzzy relation equations over complete infinitely distributive lattices, and put the question: is the minimal solution found by the procedure unique or not? In this paper, we give a negative answer to the question and make some further remarks. We not only give a necessary and sufficient condition for the uniqueness of such minimal solutions, but also characterize the existence of the least solution and a unique solution of a system of fuzzy relation equations over complete infinitely distributive lattices.
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We are grateful to the editor and the anonymous referees for their valuable comments and suggestions which helped to improve the paper.
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Communicated by A. Di Nola.
Supported by National Natural Science Foundation of China (No. 11171242) and Scientific Reserch Fund of Sichuan Provincial Education Department (No. 13ZB0108).
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Sun, F., Qu, Xb. & Wang, Xp. Remarks on minimal solutions of a system of fuzzy relation equations over complete infinitely distributive lattices. Soft Comput 20, 423–428 (2016). https://doi.org/10.1007/s00500-015-1771-9
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DOI: https://doi.org/10.1007/s00500-015-1771-9