Remarks on minimal solutions of a system of fuzzy relation equations over complete infinitely distributive lattices

摘要

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.