Standard theories of fuzzy sets with the law \((\mu\wedge\sigma')'=\sigma\vee(\mu'\wedge\sigma')\)
Usually, a lattice-valued algebra of fuzzy sets is not Boolean [cf., however, \textit{J. G. Brown}, Inf. Control 18, 32--39 (1971; Zbl 0217.01403)]. Thus, we use partial structures as De Morgan lattices or incline algebras [cf. \textit{K. H. Kim} and \textit{F. W. Roush}, Linear Algebra Appl. 379, 457--473 (2004; Zbl 1044.15009)]. Conversely, for a particular law from propositional calculus we can characterize fuzzy set operations fulfilling this law [cf., e.g., \textit{M. Mas, G. Mayor} and \textit{J. Torrens}, Fuzzy Sets Syst. 126, 207--218 (2002; Zbl 0996.03038)]. This paper characterizes the propositional property of the title using the product triangular norm \(T_P\) and the Łukasiewicz triangular conorm \(S_L\) [cf. \textit{E. P. Klement, R. Mesiar} and \textit{E. Pap}, Triangular norms, Kluwer, Dordrecht (2000; Zbl 0972.03002)]. The authors underline that the obtained algebra of fuzzy sets fails to satisfy the De Morgan laws [for a similar effect, cf. \textit{W. Homenda}, ``Weak axiomatization of balanced operators'', in: Soft computing foundations and theoretical aspects, EXIT, Warszawa, 263--276 (2004)].
scientific article; zbMATH DE number 2102560