Embeddings into the generic poset

TL;DR

This paper demonstrates that any countably infinite poset can be embedded into the generic poset in a way that automorphisms extend uniquely, highlighting the universality and automorphism extension properties of the generic poset.

Contribution

It establishes the existence of embeddings of all countably infinite posets into the generic poset with automorphism extension, revealing new structural insights.

Findings

Every countably infinite poset embeds into the generic poset with coinfinite image.

Automorphisms of the embedded poset extend uniquely to automorphisms of the generic poset.

The generic poset exhibits a universal and highly symmetric structure.

Abstract

Let M be the generic poset, defined as the Fra\"iss\'e limit of the class of finite posets. We show that every countably infinite poset A can be embedded with coinfinite image into M so that each automorphism of the image of A extends uniquely to an automorphism of M.