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. 1. Mathematical Morphology Mathematical morphology is a method of non-linear signal processing using simple set-theoretic operations, which has the feasibility of extracting the characteristic properties of shapes [1, 2]. In this paper we will adopt the formulation thereof generalized on lattices [3–7]. We identify a binary relation and the correspondence from to . Namely, for . We call the relation with and exchanged, the transpose of and denote it by . 1.1. Dilation and Erosion Let , be partially ordered sets. If for any family of which has a supremum in , the image has the supremum in and holds, then we call the mapping a dilation from to . Similarly, by changing supremum by infimum, we may introduce an erosion. We call dilation and erosion morphological operations. For two elements of , we have , , the morphological operations are monotone. Example 1.1 (morphology of set lattices [7]). Given sets and , consider the lattices of their power sets , . Let be a binary relation in . Then the mappings and defined by are a dilation and an erosion, respectively. From the transpose we may similarly define the dilation and erosion , . The importance of this example lies in the fact that all morphological operations between set lattices are expressed in this form, whence it follows