# Pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes

## TL;DR

This paper introduces the concepts of pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, establishing their equivalence and extending existing theories in combinatorial commutative algebra.

## Contribution

It defines new classes of monomial ideals and multicomplexes, proving their equivalence and generalizing prior results in the field.

## Key findings

- A multicomplex is $k$-decomposable iff its associated monomial ideal is pretty $k$-clean.
- A monomial ideal is pretty $k$-clean iff its polarization is $k$-clean.
- The results extend previous work by Herzog-Popescu, Soleyman Jahan, and the author.

## Abstract

We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $\Gamma$ is $k$-decomposable if and only if its associated monomial ideal $I(\Gamma)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author.