Mathematics > Quantum Algebra
[Submitted on 15 Feb 2025
]
Title: Classification of states on certain orthomodular structures
Title: 状态在某些正交模结构上的分类
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.
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