Operads of finite posets

TL;DR

This paper introduces four natural operad structures on the vector space of finite posets, revealing isomorphisms and simplified versions that relate to well-known operads like NAP and pre-Lie.

Contribution

It defines and analyzes four operad structures on finite posets, highlighting their relationships and simplifications, which was not previously established.

Findings

Two of the operads are isomorphic.

Three operads are set-theoretical.

The structures relate to NAP and pre-Lie operads.

Abstract

We describe four natural operad structures on the vector space generated by isomorphism classes of finite posets. The three last ones are set-theoretical and can be seen as a simplified version of the first, the same way the NAP operad behaves with respect to the pre-Lie operad. Moreover the two first ones are isomorphic.