Integral operators on lattices

TL;DR

This paper introduces the concept of integral operators on lattices, generalizing classical integral operators, and explores their properties, classifications, and related algebraic structures.

Contribution

It initiates the study of Rota-Baxter operators on lattices, characterizes lattice properties via these operators, and classifies their isomorphism classes on common lattices.

Findings

Characterization of lattice properties through integral operators

Classification of integral operators on specific lattices

Development of algebraic structures from differential and integral operators

Abstract

As an abstraction and generalization of the integral operator in analysis, integral operators (known as Rota-Baxter operators of weight zero) on associative algebras and Lie algebras have played an important role in mathematics and physics. This paper initiates the study of integral operators on lattices and the resulting Rota-Baxter lattices (of weight zero). We show that properties of lattices can be characterized in terms of their integral operators. We also display a large number of integral operators on any given lattice and classify the isomorphism classes of integral operators on some common classes of lattices. We further investigate structures on semirings derived from differential and integral operators on lattices.