Operads from posets and Koszul duality

Overview & Synopsis

We introduce a functor As from the category of posets to the category of non-symmetric binary and quadratic operads, establishing a new connection between these two categories. Each operad obtained by the construction As provides a generalization of the associative operad because all of its generating operations are associative. This construction has a very singular property: the operads obtained from As are almost never basic. Besides, the properties of the obtained operads, such as Koszulity, basicity, associative elements, realization , and dimensions, depend on combinatorial properties of the starting posets. Among others, we show that the property of being a forest for the Hasse diagram of the starting poset implies that the obtained operad is Koszul. Moreover, we show that the construction As restricted to a certain family of posets with Hasse diagrams satisfying some combinatorial properties is closed under Koszul duality.