Doubling bialgebras of finite topologies

TL;DR

This paper introduces two new bimonoid structures on species of finite topologies, exploring their interactions and dual associative products to deepen understanding of their algebraic properties.

Contribution

It defines a doubling of the species of finite topologies and constructs bimonoid structures with cointeraction, advancing the algebraic framework of finite topological spaces.

Findings

Established bimonoid structures on doubled species

Described cointeraction between the bimonoid structures

Analyzed dual associative products related to the structures

Abstract

The species of finite topological spaces admits two graded bimonoid structures, recently defined by F. Fauvet, L. Foissy, and the second author. In this article, we define a doubling of this species in two different ways. We build a bimonoid structure on each of these species and describe a cointeraction between them. We also investigate two related associative products obtained by dualisation.