Galois connections for incidence Hopf algebras of partially ordered sets (Q1977175)

Given the category of posets and order-preserving maps, it is the case that some associated categories, as, e.g., incidence algebras of locally finite posets over fields \(k\) with \(k\)-algebra homomorphisms, cannot be functorially connected in a very nice way. One cure is to restrict the order-preserving maps to a more special type, e.g., order-homomorphisms which preserve strict incomparability also. Another cure is to ``enrich'' the assorted category to a new one which does permit such a functorial connection. In this very nice paper the second approach is shown to be successful in that the category of incidence algebras is expanded to the category of bi-modules over such algebras with associated bi-module homomorphisms and dually to incidence Hopf algebras as well. Using this method a new argument is provided for the Rota formula describing the relationship between the Moebius functions of two (locally finite) posets related via a Galois connection (op-maps \(f: P\to Q\), \(g: Q\to P\), \(f(x)\leq w\) iff \(x\leq g(w)\)) as essentailly the associative law \(\mu_P\bullet(\xi\bullet \mu_Q)= (\mu_P\bullet \xi)\bullet \mu_Q\) on the associated bi-module \(M_{f,g}\), with an associated universal multiplicative function and Rota formula for the Hopf algebra case which reduces to the classical one when applying \(\xi_P\otimes \xi_Q\) to the general formula. The notion of Cartesian poset emphasized here is seen to be useful in that many posets are Cartesian and in that (locally finite) posets have Cartesian ``envelopes'' into which they may be optimally embedded.