Complete relations on fuzzy complete lattices. (English) Zbl 1426.06004
From the text: We introduce a notion of complete binary fuzzy relation on complete fuzzy lattice (completely lattice fuzzy ordered set). The notion leads in ordinary (crisp) case to the classical notion of complete relation on complete lattice, but reformulated in terms of the theory of power structures. We prove some basic properties of power structures of fuzzy ordered sets. In the main part of the paper, we define complete fuzzy binary relations and complete fuzzy tolerances and investigate their properties. Our main results are covered by Theorem 15 and 16. We show that a fuzzy complete lattice can be factorized by means of a complete fuzzy tolerance and that there is a naturally-defined structure of fuzzy complete lattice on the factor set. This result corresponds to the known result from the ordinary case (as in [G. Czedli, Acta Sci. Math. 44, 35–42 (1982; Zbl 0484.06010); R. Wille, in: Contributions to general algebra 3, Proc. Conf., Vienna 1984, 397–415 (1985; Zbl 0563.06006)]). In addition, we found an isomorphism between the fuzzy ordered sets of all complete fuzzy tolerances and extensive isotone fuzzy Galois connections on a fuzzy complete lattice. This result is useful for testing fuzzy tolerances for completeness. One consequence of our results is that the condition of compatibility from the definition of complete relation on a completely lattice \(\mathbf{L}\)-ordered set (Sec. 4) is redundant for \(\mathbf{L}\)-tolerances. This leads to an open problem, namely, whether the condition of compatibility follows from the other conditions of the definition.
Reviewer: Tomasz Kubiak (Poznan)
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