A Preliminary Report on Scarf Complexes of Posets

TL;DR

This paper generalizes the concept of Scarf complexes from lattices to more abstract posets, removing the reliance on integer lattices while preserving their core properties.

Contribution

It introduces a new framework for defining Scarf complexes on posets, extending their applicability beyond lattice structures.

Findings

Established a generalized definition of Scarf complexes for posets

Demonstrated the preservation of key properties without lattice dependence

Provided initial theoretical insights into the structure of these complexes

Abstract

The Scarf complex for lattices is well understood and utilized. In 2014, the author expanded the use of the term Scarf complex to encompass an infinite set in $\mathbb{Z}^n$ that was generated via an action of a lattice $\Lambda\in\mathbb{Z}^n$ upon a finite subset $A\subset\mathbb{N}^n$. This paper aims to further generalize the use of Scarf complex by removing the integer lattice completely while maintaining the essence of the Scarf complex.