We exhibit an internal coproduct on the Hopf algebra of finite topologies recently defined by the second author, C. Malvenuto and F. Patras, dual to the composition of "quasiormoulds", which are the natural version of J. Ecalle's moulds in this setting. All these results are displayed in the linear species formalism.A finite topological space with 10 elements and 4 equivalence classes Acknowlegdements: This work is supported by Agence Nationale de la Recherche, projet CARMA (Combinatoire Algébrique, Résurgence, Moules et Applications, ANR-12-BS01-0017). Refinement and quotient topologiesLet T and T ′ be two topologies on a finite set X. We say that T ′ is finer than T , and we write T ′ ≺ T , when any open subset for T is an open subset for T ′ . This is equivalent to the fact that for any x, y ∈ X, x ≤ T ′ y ⇒ x ≤ T y.The quotient T /T ′ of two topologies T and T ′ with T ′ ≺ T is defined as follows: the associated quasi-order ≤ T /T ′ is the transitive closure of the relation R defined by:(4)xRy ⇐⇒ (x ≤ T y or y ≤ T ′ x).Note that, contrarily to what is usually meant by "quotient topology", T /T ′ is a topology on the same finite space X than the one on which T and T ′ are given. The definitions immediately yield compatibility of the quotient with the involution, i.e.(5) T /T ′ = T T ′ .Examples:
An internal coproduct is described, which is compatible with Hoffman's quasishuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with Ecalle's mould calculus, i.e., mould composition and contracting arborification is precised.1 Note that the comould C is chosen to be a monoid antimorphism in [15]. KURUSCH EBRAHIMI-FARD, FRÉDÉRIC FAUVET, AND DOMINIQUE MANCHONThe algebra k Ω is the dual of the coalgebra k Ω of noncommutative polynomials with variables in Ω, endowed with the deconcatenation coproduct:Moreover, a mould gives rise, by linear extension, to a unique linear form on k Ω . The identification of the vector space of moulds with k Ω is achieved through the map:which associates to each mould M its corresponding word series W M ∈ k Ω . It is wellknown [26] that H Ω = (k Ω , ∐ ∐ -, ∆) is a commutative Hopf algebra, where ∐ ∐ -is Hoffman's quasi-shuffle product, recursively defined by ω∐∐ -1 = 1∐∐ω = ω (here 1 stands for the empty word) and:. Here a and b are letters in Ω and ω ′ , ω ′′ are words in Ω * . The notation [a + b] stands for the internal sum of the two letters in the commutative semigroup Ω. The Hopf algebra H Ω is (Ω ⊔ {0})-graded by the weight defined by ||1|| := 0 and: ||ω|| := [ω 1 + · · · + ω ℓ ] ∈ Ω for a word ω = ω 1 · · · ω ℓ ∈ Ω * of length |ω| := ℓ. The mould product × of [15] is hence obtained by dualizing the deconcatenation coproduct ∆, and thus reflects the noncommutative concatenation product in k Ω , namely:
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