×

Perspectivity and congruence in partial abelian semigroups. (English) Zbl 0938.03094

In the paper, the notion of a partial abelian semigroup is introduced as a generalization of various structures considered in the literature on quantum logics: orthomodular posets, orthoalgebras, D-posets (or effect algebras), and their respective non-unital variants. A partial abelian semigroup (PAS) is defined as a structure \((L,\perp , \oplus)\), where \(\perp \) is a binary relation on \(L\) and \(\oplus \) is a partially defined binary operation with domain \(\perp \), which is commutative and associative in the restricted sense: if one side of the relevant equality exists, so does the other and equality holds. To any subset \(M\) of a partial abelian semigroup \(L\), a relation of perspectivity \(\sim _M\) is introduced, where \(a\sim _M b\) if there exists some element \(c\in L\) such that \(a\oplus c\) and \(b\oplus c\) both exist and belong to \(M\). If \(\sim _M\) is a faithful congruence, that is, \(\sim _M\) is an equivalence relation such that for all \(a,b,c\in L\), \(a\sim _M b\) and \(c\perp a\) imply \(c\perp b\), \(M\) is called an algebraic set. The theory of pairs \((L,M)\), where \(L\) is a PAS and \(M\) is a fixed algebraic subset provides a natural generalization of the theory of manuals.

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] DINGES H.: Zur Algebra der Masstheorie. Bull. Greek Math. Soc. 19 (1978), 25-97. · Zbl 0437.28005
[2] BENNETT M. K.-FOULIS D. J.: Tensor products of orthoalgebras. Order 10 (1993), 271-282. · Zbl 0798.06015 · doi:10.1007/BF01110548
[3] DVUREČENSKIJ A.: Tensor products of difference posets. Trans. Amer. Math. Soc. 347 (1995), 1043-1057. · Zbl 0859.03031 · doi:10.2307/2154888
[4] DVUREČENSKIJ A.-PULMANNOVÁ S.: D-test spaces and difference posets. Rep. Math. Phus. 34 (1994), 151-170. · Zbl 0820.03038 · doi:10.1016/0034-4877(94)90034-5
[5] FOULIS D. J.-BENNETT M. K.: Effect algebras and unsharp quantum logics. Found. Phus. 24 (1994), 1325-1346. · Zbl 1213.06004 · doi:10.1007/BF02283036
[6] FOULIS D. J.-GREECHIE R. J.-RÜTTIMANN G. T.: Filters and supports in orthoalgebras. Internat. J. Theoret. Phus. 31 (1992), 789-807. · Zbl 0764.03026 · doi:10.1007/BF00678545
[7] GIUNTINI R.-GREULING H.: Toward a formal language for unsharp properties. Found. Phus. 19 (1989), 931-945.
[8] GREECHIE R. J.-FOULIS D. J.-PULMANNOVÁ S.: The center of an effect algebra. Order 12 (1995), 91-106. · Zbl 0846.03031 · doi:10.1007/BF01108592
[9] GUDDER S.: Quantum Probability. Academic Press, San Diego, 1988. · Zbl 0653.60004 · doi:10.1007/BF00670748
[10] HEDLÍKOVÁ J.-PULMANNOVÁ S.: Generalized difference posets and orthoalgebras. Acta Math. Univ. Comenian. LXV (1996), 247-279. · Zbl 0922.06002
[11] KOPKA F.-CHOVANEC F.: D-posets. Math. Slovaca 44 (1994), 21-34. · Zbl 0789.03048
[12] MAYET-IPPOLITO A.: Generalized orthomodular posets. Demonstratio Math. 24 (1991), 263-274. · Zbl 0755.06006
[13] PULMANNOVÁ S.-WILCE A.: Representations of D-posets. Internat. J. Theoret. Phys. 34 (1995), 1689-1696. · Zbl 0841.03035 · doi:10.1007/BF00676282
[14] WILCE A.: Partial abelian semigroups. Internat. J. Theoret. Phys. 34 (1995), 1807-1812. · Zbl 0839.03047 · doi:10.1007/BF00676295
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.