States on semi-divisible generalized residuated lattices reduce to states on MV-algebras
The authors continue their study of the set MV\((L)\) of complemented elements of a residuated lattice \(L\) [Soft Comput. 12, No.~4, 353--357 (2008; Zbl 1138.06006)]. Such structures are related to mathematical fuzzy logic as well as to extended probability theory. In the present paper they deal with good generalized (or pseudo) residuated lattices (the underlying monoidal operation need not be commutative, \(x^{-\sim} = x^{\sim -}\), and then MV\((L)\) = \(\{x\in L: x^{-\sim} = x^{\sim -} = x\}\) = \(\{x^-: x\in L\}\) = \(\{x^\sim: x\in L\}\)). Theorem 12 states that those good residuated lattices whose complemented elements induce a pseudo-MV-algebra are exactly semi-divisible (a condition weaker that divisibility) generalized residual lattices. Further, they study maximal filters on a semi-divisible generalized residuated lattice \(L\) and their relationships to maximal filters on MV\((L)\). Finally, they show (as stated in the title) that Riečan and Bosbach states reduce to states on MV-algebras.
scientific article; zbMATH DE number 5599504
scientific article