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Keywords:
quantum logic; s-map; fuzzy relations
quantum logic; s-map; fuzzy relations
Summary:
A quantum logic is one of possible mathematical models for non-compatible random events. In this work we solve a problem proposed at the conference FSTA 2006. Namely, it is proved that s-maps are symmetric fuzzy relations on a set of all symmetric and idempotent binary matrices. Consequently s-maps are not antisymmetric fuzzy relations. This paper also explores other properties of s-maps, j-maps and d-maps. Specifically, it is proved that s-maps are neither reflexive, nor irreflexive, and nor transitive, j-maps have the same properties as s-maps and d-maps are reflexive and not transitive.
References:
[1] Al-Adilee, M. A., Nánásiová, O.: Copula and s-map on a quantum logic. Inform. Sci. 24 (2009), 4199-4207. DOI | MR 2722377
[2] Lostaks, A.: L-sets and L-valued structures. Nr. 2. University of Latvia, 2003.
[3] Mesiar, R., Klement, E. P.: Open problems posed at the eighth international conference on fuzzy set theory and applications. Kybernetika 42 (2006), 2, 225-235. MR 2241786
[4] Nánásiová, O., Valášková, Á.: Maps on a quantum logic. Soft Computing 14 (2010), 1047-1052. DOI | MR 2722377
[5] Nánásiová, 0., Pulmannová, S.: S-map and tracial states. Inform. Sci. 5 (2009), 515-520. DOI | MR 2490191
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