TL;DR
This paper introduces a new class of operads called $k$-signaletic operads, explores their Koszul duals, and demonstrates their actions on combinatorial structures like multipermutations and multiposets, expanding operad theory.
Contribution
It generalizes existing operads to a new family called $k$-signaletic operads and studies their duals and combinatorial actions, providing new models and operadic constructions.
Findings
Koszul duals of $k$-signaletic operads act freely on multipermutations
New operads are obtained as Manin powers of the $L$-operad
Provides combinatorial models for these generalized operads
Abstract
We introduce -signaletic operads and their Koszul duals, generalizing the dendriform, diassociative and duplicial operads (which correspond to the case). We show that the Koszul duals of the -signaletic operads act on multipermutations and that the resulting algebras are free, thus providing combinatorial models for these operads. Finally, motivated by these actions on multipermutations, we introduce similar operations on multiposets which yield yet another relevant operad obtained as Manin powers of the -operad.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology