The rational numbers have been used to measure quantities since ancient times; however, their implementation in computer languages raises a significant problem: zero has no inverse. To address this issue, J. Bergstra and J. Tucker introduced an algebraic structure called a meadow, which allows for the inversion of zero.

In this talk, I will introduce meadows and their various classes, demonstrating that they correspond to labelled lattices, where the rings label the vertices . We will explore how concepts from ring theory, such as Artinian rings and decomposition theorems, can be adapted to this new context. Finally, I will present a connection between meadows and sheaves over a topological space, highlighting the implications of this relationship.

References

- J. Dias, and B. Dinis. ”Strolling through common meadows.” Communications in Algebra, 1–28.

- J. Dias, and B. Dinis. ”Towards an enumeration of finite common meadows.” International Journal of Algebra and Computation, 1-19

- J. Dias, B. Dinis and P. Marques . Bridging Meadows and Sheaves. arXiv:2410.05921