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Extending coarse-grained measures
No full textIn this note it is shown that extensions of measures from a finite coarse-grained (and circle coarse-grained) space to the entire power algebra can be obtained by the HornTarski extension technique in a purely combinatorial manner (without using techniques of linear spaces). Pure measures are characterized, consequences for circulant matrices are obtained
Extending states on finite concrete logics
No full textIn this note we collect several observations on state extensions. They may be instrumental to anyone who pursues the theory of quantum logics. In particular, we find out when extensions (resp. signed extensions) exist in the “concrete" concrete logic of all even-element subsets of an even-element set. We also mildly add to the study of difference-closed logics as initiated in Ovchinnikov (1999) by finding an extension theorem for subadditive states. Our results suplement the research previously carried on by De Simone (2000), Gudder (1979), Gudder and Marchand (1980), Navara and Pt ́ak (1983), Navara (1989), Ovchinnikov (1999), Ptak (2000), Ptak and Pulmannova (1991), Prather (1980), Sherstnev (1968), Sultanbekov (1992), and Svozil (1998)
Relatively additive states on quantum logics
Full text linksummary:In this paper we carry on the investigation of partially additive states on quantum logics (see [2], [5], [7], [8], [11], [12], [15], [18], etc.). We study a variant of weak states — the states which are additive with respect to a given Boolean subalgebra. In the first result we show that there are many quantum logics which do not possess any 2-additive central states (any logic possesses an abundance of 1-additive central state — see [12]). In the second result we construct a finite 3-homogeneous quantum logic which does not possess any two-valued 1-additive state with respect to a given Boolean subalgebra. This result strengthens Theorem 2 of [5] and presents a rather advanced example in the orthomodular combinatorics (see also [9], [13], [4], [6], [16], etc.). In the rest we show that Greechie logics allow for -additive three-valued states, and in case of Greechie lattices we show that one can even construct many -additive two-valued states. Some open questions are posed, too
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