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We elaborate on the notion of a filtration of an operad defined in terms of alattice-valued operad serving as an indexing object. That covers ordinaryinteger-indexed filtrations of associative algebras and operads as a specialcase, yet the notion appears to be natural enough to encompass examples ofother kind as well. The characteristic property of lattice operads is that of acertain distributivity of partial compositions with respect to meets and joins.We observe that some well-known families of lattices of combinatorial origin,such as Tamari lattices, assemble to operads subject to this particularproperty. Other examples include an operad of integer paritions supported onYoung's lattice, operads of integer compositions of types $A, B$ and $D$, whichwe relate to operads of regular polytopes.

Lattice operads and operad filtrations