On the (semi)lattices induced by continuous reducibilities

TL;DR

This paper explores the order-theoretic properties of continuous reducibilities in computable analysis, introducing suprema for various variants and examining their interaction with characteristic numbers.

Contribution

It introduces and analyzes the suprema of different variants of continuous reducibilities, advancing the understanding of their order-theoretic structure.

Findings

Suprema for variants of continuous reducibilities are established.

Suprema commute with several characteristic numbers.

The study enhances the theoretical framework of continuous reducibilities.

Abstract

Continuous reducibilities are a proven tool in computable analysis, and have applications in other fields such as constructive mathematics or reverse mathematics. We study the order-theoretic properties of several variants of the two most important definitions, and especially introduce suprema for them. The suprema are shown to commutate with several characteristic numbers.