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Difference operators on lattices

Aiping Gan, Li Guo

Abstract

A differential operator of weight $λ$ is the algebraic abstraction of the difference quotient $d_λ(f)(x):=\big(f(x+λ)-f(x)\big)/λ$, including both the derivation as $λ$ approaches to $0$ and the difference operator when $λ=1$. Correspondingly, differential algebra of weight $λ$ 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.

Difference operators on lattices

Abstract

A differential operator of weight is the algebraic abstraction of the difference quotient , including both the derivation as approaches to and the difference operator when . Correspondingly, differential algebra of weight 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 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.
Paper Structure (18 sections, 36 theorems, 34 equations)

This paper contains 18 sections, 36 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Derivations and lattices
  3. Difference operators and lattices
  4. Layout of the paper
  5. Differential operators with weights and difference operators
  6. Differential operators of weight $-1$ and derivations
  7. Properties and first examples of difference operators
  8. Derived difference operators
  9. Difference operators on finite chains
  10. Difference operators on quasi-antichains
  11. Families of difference operators on a quasi-antichain
  12. Notations and the classification of difference operators when $d(0)=0$
  13. Families of difference operators when $d(0)=1$
  14. Families of difference operators when $d(0)\neq 0,1$
  15. The classification and enumeration of difference operators
  16. ...and 3 more sections

Key Result

Lemma 2.1

Xin1 Let $L$ be a lattice and $d\in \mathrm{DO}(L)$. Then the following statements hold.

Theorems & Definitions (73)

  • Definition 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • Proposition 2.6
  • ...and 63 more