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On differential lattices

  • Foundation, algebraic, and analytical methods in soft computing
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Abstract

This paper studies the differential lattice, defined to be a lattice L equipped with a map \(d:L\rightarrow L\) that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications of differential lattices are obtained for some basic lattices. Several families of derivations on a lattice are explicitly constructed, giving realizations of the lattice as lattices of derivations. Derivations on a finite distributive lattice are shown to have a natural structure of lattice. Moreover, derivations on a complete infinitely distributive lattice form a complete lattice. For a general lattice, it is conjectured that its poset of derivations is a lattice that uniquely determines the given lattice.

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Funding

This work is supported by National Natural Science Foundation of China (Grant Nos. 12171022 and 11801239). The authors thank the referee for helpful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, Jiangxi, People’s Republic of China

    Aiping Gan

  2. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, USA

    Li Guo

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  1. Aiping Gan
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  2. Li Guo
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Correspondence to Li Guo.

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Gan, A., Guo, L. On differential lattices. Soft Comput 26, 7043–7058 (2022). https://doi.org/10.1007/s00500-022-07101-z

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