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

TL;DR

This paper extends the concepts of states and measures to pseudo-BCK algebras, characterizes extremal states, and explores their relationships with de Finetti maps and Borel states, generalizing previous results.

Contribution

It introduces new notions of states on pseudo-BCK algebras, characterizes extremal states, and links these to de Finetti maps and Borel states, broadening the theoretical framework.

Findings

States coincide with Bosbach states under certain conditions

Quotient pseudo-BCK algebras embed into archimedean $ ext{l}$-groups

Relationships established between de Finetti maps, Bosbach states, and Borel states

Abstract

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 $\ell$-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.