A Comparison of Implications in Orthomodular Quantum Logic—Morphological Analysis of Quantum Logic

Morphological operators are generalized to lattices as adjunction pairs (Serra, 1984; Ronse, 1990; Heijmans and Ronse, 1990; Heijmans, 1994). In particular, morphology for set lattices is applied to analyze logics through Kripke semantics (Bloch, 2002; Fujio and Bloch, 2004; Fujio, 2006). For example, a pair of morphological operators as an adjunction gives rise to a temporalization of normal modal logic (Fujio and Bloch, 2004; Fujio, 2006). Also, constructions of models for intuitionistic logic or linear logics can be described in terms of morphological interior and/or closure operators (Fujio and Bloch, 2004). This shows that morphological analysis can be applied to various non-classical logics. On the other hand, quantum logics are algebraically formalized as orhomodular or modular ortho-complemented lattices (Birkhoff and von Neumann, 1936; Maeda, 1980; Chiara and Giuntini, 2002), and shown to allow Kripke semantics (Chiara and Giuntini, 2002). This suggests the possibility of morphological analysis for quantum logics. In this article, to show an efficiency of morphological analysis for quantum logic, we consider the implication problem in quantum logics (Chiara and Giuntini, 2002). We will give a comparison of the 5 polynomial implication connectives available in quantum logics.