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 .
To save content items to your Kindle, first ensure email@example.com
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.
In this chapter, we study further the Hadamard product on species. The Hadamard product of two free monoids is again free. Similarly, the Hadamard product of two free commutative monoids is again free commutative. In either case, we give an explicit formula for a basis of the Hadamard product in terms of bases of its two factors. It involves the meet operation on faces and flats, respectively. We also show that the Hadamard product of bimonoids is free as a monoid if one of two factors is free as a monoid. We study in detail the Hadamard product of the free bimonoid on a comonoid with the cofree bimonoid on a monoid. It is neither commutative nor cocommutative, so Borel-Hopf does not apply. This bimonoid is both free and cofree. Interestingly, we prove this using Loday-Ronco (which is a theorem about 0-bimonoids). We also give a cancelation-free formula for its antipode. We give an explicit description of its primitive part, and more generally, its primitive filtration. An illustrative example of this construction is the bimonoid of pairs of chambers. We give a parallel discussion for a commutative counterpart where we take the Hadamard product of the free commutative bimonoid on a cocommutative comonoid with the cofree cocommutative bimonoid on a commutative monoid. (Since this bimonoid is bicommutative, it can also be tackled using Leray-Samelson.)
Email your librarian or administrator to recommend adding this to your organisation's collection.