Generalized Bosbach and Riečan states based on relative negations in residuated lattices (Q427915)

A state, an analogue of a probability measure, on algebraic structures like MV-algebras \(M\) is an additive functional \(s:M \to [0,1]\) such that \(s(1)=1\) and \(s(a\oplus b)= s(a)+s(b)\) whenever \(a\odot b =0.\) In the same way we can define states on pseudo MV-algebras. On BL-algebras we do not have the operation \(\oplus\), and moreover, if we have pseudo BL-algebras, the situation is even more complicated. Therefore, two notions of states were introduced: Riečan states, which imitate the situation for MV-algebras, and Bosbach states. If, for example, a pseudo BL-algebra is good, i.e. two negations commute, these two notions are equivalent.NEWLINENEWLINEThe paper under review continues the study of states in residuated lattices. The authors study two notions of generalized Bosbach states of type I and type II, respectively, and a Riečan state, and they show some relations between them. In addition, several necessary and sufficient conditions for the set of all relatively regular elements are presented, which gives a Glivenko-type result.