# 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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