On the fuzziness measure and negation in totally ordered lattices (Q1087529)

\textit{R. R. Yager} [Inf. Control 44, 236--260 (1980; Zbl 0429.04008)] considered (a degree of) fuzziness to be defined in a lattice as a lack of distinction of an element and its ``negation''. The authors continue this approach. Starting from any chain \(L\) with an order-reversing involutionary unary operation they get a new chain \(L^{\wedge}\) with \(| L| =| L^{\wedge}|\) by introducing a suitable fuzziness ordering. By an order-reversing lattice morphism, \(L^{\wedge}\) can be embedded into the lattice of negate pairs. Finally, valuations on \(L\) lead to valuations on \(L^{\wedge}\), and for \(L={\mathcal F}(X)\) such valuations give measures of fuzziness for fuzzy subsets of \(X\).