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

Abstract

Abstract. In this paper, we extend the notions of states and measures presented in [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 [12] coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. 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 J. Kühr and D. Mundici from [28], we prove a relationship between de Finetti maps, Bosbach states and Borel states.

Applications of states to BI-algebras

“…It is known that the notion of state was firstly defined on an MV-algebra by Kõpka and Chovane in [32], and then has been studied and applied to other algebraic structures, since they have important roles in studying logical algebras (see, for instance, Borzooei et al [3,4,5], Buşneag [8,9], Ciungu [13,14,15], Ciungu and Dvurečenskij [17], Ciungu et al [16,18], Chen and Dudek [10], Cheng et al [11], Di Nola and Dvurečenskij [21], Di Nola et al [22], Dvurečenskij and Zahiri https://doi.org/10.52547/HATEF.JAHLA. 3.3.4 (vi) if x * y = z, then z * y = z and y * z = y, (vii) if (x * y) * (z * u) = (x * z) * (y * u), then X = {0}.…”

State BCK-algebras and state-morphism BCK-algebras

“…Their method is applicable to other structures besides MV-algebras. Measures on pseudo BCK-algebras were studied in [7].…”