# A new tool to study the fixed point property of finite posets

## TL;DR

This paper introduces a new topological tool for analyzing the fixed point property in finite posets, using a construction that induces an endofunctor in the homotopy category of finite T0-spaces.

## Contribution

It presents a novel topological construction that provides a new method to study fixed point properties of finite posets, with applications demonstrated through examples.

## Key findings

- The construction induces an endofunctor in the homotopy category of finite T0-spaces.
- The tool helps analyze fixed point properties using topological methods.
- Several applications illustrate the effectiveness of the new construction.

## Abstract

We develop a novel tool to study the fixed point property of finite posets using a topological approach. Our tool is a construction which turns out to induce an endofunctor of the homotopy category of finite $T_0$--spaces. We study many properties of this construction and give several examples of application.