Lattice operads and operad filtrations

TL;DR

This paper introduces a generalized notion of operad filtrations indexed by lattices, unifying various known examples and exploring their combinatorial and algebraic properties.

Contribution

It defines lattice-valued operad filtrations, demonstrating their naturalness and connecting them to well-known combinatorial lattices and operad structures.

Findings

Lattice operads exhibit distributivity of partial compositions.

Tamari lattices form operads with this property.

Operads of integer partitions and compositions relate to polytopes and permutation orders.

Abstract

We elaborate on the notion of a filtration of an operad defined in terms of a lattice-valued operad serving as an indexing object. That covers ordinary integer-indexed filtrations of associative algebras and operads as a special case, yet the notion appears to be natural enough to encompass examples of other kind as well. The characteristic property of lattice operads is that of a certain 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 particular property. Other examples include an operad of integer paritions supported on Young's lattice, operads of integer compositions of types $A, B$ and $D$, which we relate to operads of regular polytopes. We discuss the partial compatibility of the weak order on the symmetric group with the structure of the permutations operad.