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GALOIS CONNECTIONS FOR INCIDENCE HOPF ALGEBRAS OF PARTIALLY ORDERED SETS.
 

Summary: GALOIS CONNECTIONS FOR INCIDENCE HOPF ALGEBRAS OF PARTIALLY
ORDERED SETS.
MARCELO AGUIAR AND WALTER FERRER SANTOS
Abstract. An important well­known result of Rota describes the relationship between the M¨obius
functions of two posets related by a Galois connection. We present an analogous result relating
the antipodes of the corresponding incidence Hopf algebras, from which the classical formula can
be deduced. To motivate the derivation of this more general result, we first observe that a simple
conceptual proof of Rota's classical formula can be obtained by interpreting it in terms of bimodules
over the incidence algebras. Bimodules correct the apparent lack of functoriality of incidence algebras
with respect to monotone maps. The theory of incidence Hopf algebras is reviewed from scratch, and
centered around the notion of cartesian posets. Also, the universal multiplicative function on a poset
is constructed and an analog for antipodes of the classical M¨obius inversion for formula is presented.
1. Introduction, notation and preliminaries
All posets to be considered are assumed to be locally finite. k is a fixed commutative ring, often
omitted from the notation. I P is the incidence algebra of the poset P over k:
I P = f' : P \Theta P ! k = '(x; y) = 0 if x 6Ÿ yg ;
with multiplication
(' \Lambda /)(x; y) =
X
z2P

  

Source: Aguiar, Marcelo - Department of Mathematics, Texas A&M University

 

Collections: Mathematics