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

In this paper, we extend the notions of states and measures presented in \cite{DvPu} 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 \cite{DvPu} coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. K"uhr 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 $\mathrm{\ell }$-group. Additionally, we introduce a Borel state and using results by J. K"uhr and D. Mundici from \cite{Kumu}, we prove a relationship between de Finetti maps, Bosbach states and Borel states.