# d-Complete posets: local structural axioms, properties, and equivalent definitions

## TL;DR

This paper explores the local structural axioms, properties, and equivalent definitions of d-complete posets, which generalize several classical combinatorial structures and have important algebraic and combinatorial properties.

## Contribution

It establishes the equivalence of multiple definitions of d-complete posets and summarizes their fundamental properties and background.

## Key findings

- Multiple definitions of d-complete posets are shown to be equivalent
- d-complete posets generalize rooted trees, shapes, and shifted shapes
- They possess Stanley's hook product property and jeu de taquin rectification

## Abstract

Although d-complete posets arose along the interface between algebraic combinatorics and Lie theory, they are defined using only requirements on their local structure. These posets are a mutual generalization of rooted trees, shapes, and shifted shapes. They possess Stanley's hook product property for their P-partition generating functions and Schutzenberger's well defined jeu de taquin rectification property. The original definition of d-complete poset was lengthy, but more succinct definitions were later developed. Here several definitions are shown to be equivalent. The basic properties of d-complete posets are summarized. Background and a partial bibliography for these posets is given.