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In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

In random experiments, the fact that the sets of events has a structure of a Boolean algebra, i.e. it follows the rules of classical logic, is the main hypothesis of classical probability theory. Bosbach states have been introduced on commutative and non-commutative algebras of fuzzy logics as a way of probabilistically evaluating the formulas. In this paper, we focus on the relationship between some properties of ideals and Bosbach states in the framework of commutative residuated lattices. In particular, we introduce the concept of co-kernel of a Bosbach state which is an ideal and we establish the relationship between the notion of co-kernel and the kernel. Moreover, we define and characterize maximal ideals and maximal MV-ideals in residuated lattices.

Bosbach and Riečan states on residuated lattices both are analogues of probability measures on Boolean algebras. In this paper it is proved that Riečan states on a residuated lattice is uniquely determined by its restriction on the subset of all regular elements and consequently the same holds for Bosbach states on Glivenko residuated lattices. These results indicate that for studies of Riečan and Bosbach states one can concentrate only on involutive residuated lattices.