Measures, states and de Finetti maps on pseudo-BCK algebras

In this paper, we extend the notions of states and measures presented in Dvurečenskij and Pulmannová (2000) [12] to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of Dvurečenskij and Pulmannová (2000) [12] coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by Kühr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean ℓ -group . Additionally, we introduce a Borel state and using results by Kühr and Mundici (2007) [28], we prove a relationship between de Finetti maps, Bosbach states and Borel states.