Polyocollection ideals and primary decomposition of polyomino ideals

TL;DR

This paper introduces polyocollections, a new combinatorial concept, to analyze the primary decomposition of non-prime polyomino ideals, providing explicit descriptions for certain classes like closed path polyominoes.

Contribution

It generalizes previous concepts to define polyocollections, enabling the computation of primary decompositions for broader classes of polyomino ideals.

Findings

Polyocollections generalize collections of cells and polyominoes.

Explicit minimal primary decomposition for non-prime closed path polyominoes.

Characterization of all zig-zag walks in these polyominoes.

Abstract

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.