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We now discuss a class of results related to universal constructions which we call the Hoffman-Newman-Radford (HNR) rigidity theorems. They come in different flavors which we explain one by one. In each case, the result provides explicit inverse isomorphisms between two universally constructed bimonoids. We call these the Hoffman-Newman-Radford (HNR) isomorphisms. For a cocommutative comonoid, the free bimonoid on that comonoid is isomorphic to the free bimonoid on the same comonoid but with the trivial coproduct. The product is concatenation in both, but the coproducts differ, it is dequasishuffle in the former and deshuffle in the latter. An explicit isomorphism can be constructed in either direction, one direction involves a noncommutative zeta function, while the other direction involves a noncommutative Möbius function.These are the HNR isomorphisms. There is a dual result starting with a commutative monoid.In this case, the coproduct is deconcatenation in both, but the products differ, it is quasishuffle in the former and shuffle in the latter. Interestingly, these ideas can be used to prove that noncommutative zeta functions and noncommutative Möbius functions are inverse to each other in the lune-incidence algebra. There is a commutative analogue of the above results in which the universally constructed bimonoids are bicommutative. Now the HNR isomorphisms are constructed using the zeta function and Möbius function of the poset of flats. As an application, we explain how they can be used to diagonalize the mixed distributive law for bicommutative bimonoids. There is also a q-analogue, for q not a root of unity. In this case, the HNR isomorphisms involve the two-sided q-zeta and q-Möbius functions. As an application, we explain how they can be used to study the nondegeneracy of the mixed distributive law for q-bimonoids.
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