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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 this paper, we introduce the concepts of [Formula: see text]-fold boolean ideals, [Formula: see text]-fold implicative ideals and [Formula: see text]-fold integral ideals in residuated lattices and we state and prove their properties. Several characterizations of these notions are derived and the relations between those notions are investigated. Also, we construct the correspondence between the notions of [Formula: see text]-fold ideal and [Formula: see text]-fold filter in residuated lattices.