Coordinatization of orthocomplemented and orthomodular posets
Authors:
S. P. Gudder and R. H. Schelp
Journal:
Proc. Amer. Math. Soc. 25 (1970), 229-237
MSC:
Primary 06.35
DOI:
https://doi.org/10.1090/S0002-9939-1970-0258690-3
MathSciNet review:
0258690
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Abstract | References | Similar Articles | Additional Information
Abstract: Generalizations of Baer $^{\ast }$-semigroups called partial Baer $^{\ast }$-semigroups and OM-partial Baer $^{\ast }$-semigroups are introduced. It is shown that the set of closed projections of a (OM) partial Baer $^{\ast }$-semigroup form an (orthomodular) orthocomplemented poset. Conversely (orthomodular) orthocomplemented posets are coordinatized by (OM) partial Baer $^{\ast }$-semigroups. It is shown that these coordinatizing semigroups are minimal.
- David J. Foulis, Baer $^{\ast } $-semigroups, Proc. Amer. Math. Soc. 11 (1960), 648–654. MR 125808, DOI https://doi.org/10.1090/S0002-9939-1960-0125808-6 ---, Lecture notes on orthomodular lattices, University of Massachusetts, Amherst, Mass.
- Stanley Gudder, Axiomatic quantum mechanics and generalized probability theory, Probabilistic Methods in Applied Mathematics, Vol. 2, Academic Press, New York, 1970, pp. 53–129. MR 0266552
- V. S. Varadarajan, Probability in physics and a theorem on simultaneous observability, Comm. Pure Appl. Math. 15 (1962), 189–217. MR 163616, DOI https://doi.org/10.1002/cpa.3160150207
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Additional Information
Keywords:
Partially ordered sets,
orthocomplemented posets,
orthomodular posets,
Baer <IMG WIDTH="15" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$^{\ast }$">-semigroups,
coordinatization theorems
Article copyright:
© Copyright 1970
American Mathematical Society