Bridging Meadows and Sheaves
Jo\~ao Dias, Bruno Dinis, Pedro Macias Marques

TL;DR
This paper establishes a formal connection between sheaves of rings over topological spaces and algebraic structures called common meadows, providing constructions and correspondences that unify these mathematical concepts.
Contribution
It introduces a novel equivalence between presheaves of rings and pre-meadows, and constructs a method to produce common meadows from sheaves, linking sheafification with meadow formation.
Findings
Equivalence between presheaves of rings and pre-meadows.
Construction of common meadows from sheaves of rings.
Correspondence between sheafification and meadow formation.
Abstract
We bridge sheaves of rings over a topological space with common meadows (algebraic structures where the inverse for multiplication is a total operation). More specifically, we show that the subclass of pre-meadows with , coming from the lattice of open sets of a topological space , and presheaves over are the same structure. Furthermore, we provide a construction that, given a sheaf of rings on produces a common meadow as a disjoint union of elements of the form indexed over the open subsets of . We also establish a correspondence between the process of going from a presheaf to a sheaf (called sheafification) and the process of going from a pre-meadow with to a common meadow.
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Taxonomy
TopicsLandslides and related hazards
