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L-effect Algebras

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Abstract

L-effect algebras are introduced as a class of L-algebras which specialize to all known generalizations of effect algebras with a \(\wedge \)-semilattice structure. Moreover, L-effect algebras X arise in connection with quantum sets and Frobenius algebras. The translates of X in the self-similar closure S(X) form a covering, and the structure of X is shown to be equivalent to the compatibility of overlapping translates. A second characterization represents an L-effect algebra in the spirit of closed categories. As an application, it is proved that every lattice effect algebra is an interval in a right \(\ell \)-group, the structure group of the corresponding L-algebra. A block theory for generalized lattice effect algebras, and the existence of a generalized OML as the subalgebra of sharp elements are derived from this description.

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Acknowledgements

Supported by the Natural Science Foundation of Guangdong Province, China (No. 2016A030313832), the Science and Technology Program of Guangzhou, China (No. 201607010190), and the State Scholarship Fund, China (Grant No. 201708440512).

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Authors and Affiliations

  1. Institute for Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70550, Stuttgart, Germany

    Wolfgang Rump

  2. School of Mathematical Sciences, South China Normal University, Shipai, Guangzhou, 510631, People’s Republic of China

    Xia Zhang

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  1. Wolfgang Rump
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Correspondence to Wolfgang Rump.

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Dedicated to B. V. M.

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Rump, W., Zhang, X. L-effect Algebras. Stud Logica 108, 725–750 (2020). https://doi.org/10.1007/s11225-019-09873-2

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  • DOI: https://doi.org/10.1007/s11225-019-09873-2

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