Constraints on composing product with coproduct in a bialgebra?

by HaKuNa MaTaTa   Last Updated August 01, 2020 11:20 AM - source

If $(H,\mu,\nu,\Delta,\epsilon)$ is a vector space, product, unit, coproduct and counit, and we have $x \in H$, what can be said for $\mu \circ \Delta(x)$ and $\Delta \circ \mu (x \otimes x)$?

I know that for $H$ to be a bialgebra we need $\mu, \nu$ to be coalgebra morphisms and $\Delta$,$\epsilon$ to be algebra morphisms. Any insight is appreciated. Thanks!

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