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The geometry of discrete L-algebras

  • Wolfgang Rump
From the journal Advances in Geometry

Abstract

The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined.

MSC 2010: 51A05; 51A50; 81P10; 06C15; 46C05; 03E70

Dedicated to B. V. M.


  1. Communicated by: T. Grundhöfer

Acknowledgements

The author owes thanks to an anonymous referee for careful reading of the manuscript and helpful suggestions to improve its readability for a wide audience.

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Received: 2023-01-21
Revised: 2023-05-17
Published Online: 2023-10-17
Published in Print: 2023-10-26

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