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Difference operators on lattices

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Aiping Gan, Li Guo

Rings and Algebras · Submitted 3 Feb 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Distributivity of a lattice is characterized by the existence of certain difference operators.

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arXiv ID2402.02282 (version 1)
Pith IDQL4WQFTR, Pith's permanent record of this paper and its review
CitationAiping Gan, Li Guo. "Difference operators on lattices." arXiv:2402.02282 (2024). https://pith.science/paper/QL4WQFTR
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What the abstract itself states, as probabilities: claims first 0.47 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.34 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

A differential operator of weight $\lambda$ is the algebraic abstraction of the difference quotient $d_\lambda(f)(x):=\big(f(x+\lambda)-f(x)\big)/\lambda$, including both the derivation as $\lambda$ approaches to $0$ and the difference operator when $\lambda=1$. Correspondingly, differential algebra of weight $\lambda$ extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight $-1$ on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 A differential operator of weight $\lambda$ is the algebraic abstraction of the difference quotient $d_\lambda(f)(x):=\big(f(x+\lambda)-f(x)\big)/\lambda$, including both the derivation as $\lambda$ approaches to $0$ and the difference operator when $\lambda=1$. background 1.00 0.12 0.04 asserted 1.00
s1 Correspondingly, differential algebra of weight $\lambda$ extends the well-established theories of differential algebra and difference algebra. background 1.00 0.08 0.04 asserted 1.00
s2 In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. contribution 1.00 0.81 0.07 asserted 0.94
s3 We show that differential operators of weight $-1$ on a lattice coincide with differential operators, while differential operators are special cases of difference operators. result 0.99 0.90 0.23 asserted 1.00
s4 Distributivity of a lattice is characterized by the existence of certain difference operators. result 0.99 0.85 0.27 asserted 1.00
s5 Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains. result 0.59 0.95 0.19 asserted 1.00

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Original abstract

A differential operator of weight $\lambda$ is the algebraic abstraction of the difference quotient $d_\lambda(f)(x):=\big(f(x+\lambda)-f(x)\big)/\lambda$, including both the derivation as $\lambda$ approaches to $0$ and the difference operator when $\lambda=1$. Correspondingly, differential algebra of weight $\lambda$ extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight $-1$ on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains.

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