States on Multiple-Valued Logic Algebras
In this chapter we will present the notion of state for the case of pseudo-BCK algebras. One of the main results consists of proving that any Bosbach state on a good pseudo-BCK algebra is a Riečan state, but conversely it turns out not to be true. Some conditions are given for a Riečan state on a good pseudo-BCK algebra to be a Bosbach state. In contrast to the case of pseudo-BL algebras, we show that there exist linearly ordered pseudo-BCK algebras having no Bosbach states and that there exist pseudo-BCK algebras having normal deductive systems which are maximal, but having no Bosbach states. Some specific properties of states on FL w -algebras, pseudo-MTL algebras, bounded Rℓ-monoids and subinterval algebras of pseudo-hoops are proved. A special section is dedicated to the existence of states on the residuated structures, showing that every perfect FL w -algebra admits at least a Bosbach state and every perfect pseudo-BL algebra has a unique state-morphism. Finally, we introduce the notion of a local state on a perfect pseudo-MTL algebra and we prove that every local state can be extended to a Riečan state.
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