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We introduce an operad of formal fractions, abstracted from the Mould operadsand containing both the Dendriform and the Tridendriform operads. We considerthe smallest set-operad contained in this operad and containing four specificelements of arity two, corresponding to the generators and the associativeelements of the Dendriform and Tridendriform operads. We obtain a presentationof this operad (by binary generators and quadratic relations) and an explicitcombinatorial description using a new kind of bi-colored trees. Similar resultsare also presented for related symmetric operads.

A set-operad of formal fractions and dendriform-like sub-operads