Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-22T21:43:40.171Z Has data issue: false hasContentIssue false

Chapter 10 - Characteristic operations

from Part II - Basic theory of bimonoids

Published online by Cambridge University Press:  28 February 2020

Marcelo Aguiar
Affiliation:
Cornell University, Ithaca
Swapneel Mahajan
Affiliation:
Indian Institute of Technology, Mumbai
Get access

Summary

The definition of a bimonoid in species makes use of the Tits monoid of the hyperplane arrangement. The latter is a monoid structure on the set of faces.On the other hand, there is the bimonoid of faces, which is itself built out of faces.This double occurrence of faces acquires formal meaning now. Elements of the bimonoid of faces give rise to characteristic operations on any bimonoid.This yields a morphism from the bimonoid of faces to the biconvolution bimonoid associated to the given bimonoid. Further, when the given bimonoid is commutative or cocommutative, each face-component map of this morphism is an algebra antimap or an algebra map, with the Tits product on the former and composition product on the latter. The above story has a simpler commutative analogue. Bicommutative bimonoids can be formulated using the Birkhoff monoid. The latter is a monoid structure on the set of flats. On the other hand, there is the bimonoid of flats, which is itself built out of flats. Formally, elements of the bimonoid of flats give rise to commutative characteristic operations on any bicommutative bimonoid. This yields a morphism from the bimonoid of flats to the biconvolution bimonoid associated to the given bimonoid. Further, each flat-component of this morphism is an algebra map. There are more general operations one can consider on bimonoids by working with bifaces instead of faces. We call these the two-sided characteristic operations. The role of the Tits algebra is now played by the Janus algebra. More generally, one can also consider q-bimonoids whose face-components are acted upon by the q-Janus algebra.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×