Classification of states on certain orthomodular structures
Lavinia Corina Ciungu
TL;DR
The paper investigates state notions on implicative involutive BE algebras, introducing Jauch-Piron, (P)-, (B)-, subadditive states, and valuations, and explores their interrelations. It then defines unital, full, and rich state-sets and establishes strong structural consequences: the existence of a rich or full state set forces the algebra to be an implicative-orthomodular lattice, while a rich set of (P)-states or a full set of valuations yields an implicative-Boolean algebra. Deductive-system notions (DS, o-DS, q-DS, p-DS) are developed and used to characterize IOMLs and implicative-Boolean algebras via state-based conditions. Overall, the work builds a cohesive framework linking state theory to algebraic structure and provides criteria for identifying IOMLs and implicative-Boolean algebras from state-set properties.
Abstract
We define various type of states on implicative involutive BE algebras (Jauch-Piron state, (P)-state, (B)-state, subadditive state, valuation), and we investigate the relationships between these states. Moreover, we introduce the unital, full and rich sets of states, and we prove certain properties involving these notions. In the case when an implicative involutive BE algebra possesses a rich or a full set of states, we prove that it is an implicative-orthomodular lattice. If an implicative involutive BE algebra possesses a rich set of (P)-states or a full set of valuations, then it is an implicative-Boolean algebra. Additionally, based on their deductive systems, we give characterizations of implicative-orthomodular lattices and implicative-Boolean algebras.