Mitsuhiko Fujio
International Journal of Mathematics and Mathematical Sciences 2012 2012年
[査読有り] 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. © 2012 Mitsuhiko Fujio.