# Examples of disk algebras

## TL;DR

This paper refines the multiplicative structures on various spectra like BP, BP⟨n⟩, and X(n), enabling the inheritance of circle actions in their topological Hochschild homology.

## Contribution

It introduces refined multiplicative structures on key spectra, enhancing understanding of their algebraic and topological properties.

## Key findings

- Refined multiplicative structures on BP and variants.
- Topological Hochschild homology inherits circle actions.
- Enhanced algebraic understanding of spectra.

## Abstract

We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum. Consequently, topological Hochschild homology relative to these rings inherits a circle action.