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# PolyominoIdeals -- a package to work with binomial ideals associated with collections of cells

## Description

PolyominoIdeals is a package for making several computations with the inner 2-minor ideals attached to collections of cells.

In [AAQ2012] Ayesha Asloob Qureshi establishes a connection between Combinatorial Commutiative Algebra and collection of cells, assigning to every collection of cells the binomial ideal of its inner $2$-minors.

Consider the natural partial order on $\NN^2$ and let $a,b \in \N^2$ with $a\leq b$. The set $[a, b] = \{c \in \NN^2 : a \leq c \leq b\}$ is called an interval of $\NN^2$; moreover, if $b=a+(1,1)$, then $[a,b]$ is called a cell of $\NN^2$. An interval $C=[a, b]$, where $a = (i, j)$ and $b = (k, l)$, is said to be a proper interval if $i < k$ and $j < l$. The elements $a, b$ are said the diagonal corners of $C$ and $c = (k, j)$ and $d = (i, l)$ the anti-diagonal ones. If $C$ is a cell, then $V(C)=\{a,a+(1,1),a+(0,1),a+(1,0)\}$ is the set of the corners of $C$.
To each collection of cells $\mathcal{P}$, we attach an ideal $I_{\mathcal{P}}$ as following. Let $K$ be a field and $S=K[x_a: a \in V (\mathcal{P})$, where $V (\mathcal{P})$ is the union of the vertices sets of all cells of $\mathcal{P}$. A proper interval $[a, b]$ is called an inner interval of $\mathcal{P}$ if all cells of $[a, b]$ belong to $\mathcal{P}$. The binomial $f= x_ax_b − x_c x_d$ , where $c$ and $d$ are the anti-diagonal corners of $[a, b]$, is called an inner 2-minor of $\mathcal{P}$, if $[a, b]$ is an inner interval of $\mathcal{P}$. We denote by $I_{\mathcal{P}}$ the ideal generated in $S$ by the inner 2-minors of $\mathcal{P}$ and by $K [\mathcal{P}]$ the quotient ring $S/I_{\mathcal{P}}$, called the coordinate ring of $\mathcal{P}$.
The class of ideals attached to a collection of cells includes, for example, the ideals of 2-minors of two-sided ladders, but it is much more general. Interesting classes of collections of cells are the so-called polyominoes that are well studied in various combinatorial contexts. A collection of cells $\mathcal{P}$ is called a polyomino if for any two cells $A, B \in \mathcal{P}$ there exists a sequence of cells $A=C_1,\dots, C_m=B$ of $\mathcal{P}$ such that $C_i$ and $C_{i+1}$ have an edge in common. In such a case, $I_{\mathcal{P}}$ is called polyomino ideal of $\mathcal{P}$.

The aim of this package is to provide several tools to help mathematicians in the study of polyomino ideals. Every contribution is very welcome.

Literature
• [AAQ2012] Ideals generated by 2-minors, collections of cells and stack polyominoes (A. A. Qureshi, 2012, J. Algebra).

## Version

This documentation describes version 1.1 of PolyominoIdeals.

## Source code

The source code from which this documentation is derived is in the file PolyominoIdeals.m2.

## Exports

• Functions and commands
• polyoIdeal -- Ideal of inner 2-minors of a collection of cells
• polyoMatrix -- Matrix attached to a collection of cells
• polyoToric -- Toric ideal of a polyomino
• Methods
• polyoIdeal(List) -- see polyoIdeal -- Ideal of inner 2-minors of a collection of cells
• polyoMatrix(List) -- see polyoMatrix -- Matrix attached to a collection of cells
• polyoToric(List,List) -- see polyoToric -- Toric ideal of a polyomino
• Symbols
• Field -- optional argument for polyoIdeal
• RingChoice -- optional argument for polyoIdeal
• TermOrder -- optional argument for polyoIdeal

## For the programmer

The object PolyominoIdeals is .