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Description

Given a collection of cells $\mathcal{P}$, this function returns the lattice ideal associated with $\mathcal{P}$.

Let $\mathcal{P}$ be a collection of cells. For each $a \in V(\mathcal{P})$, denote by $\mathbf{v}_a$ the vector in $\mathbb{Z}^{|V(\mathcal{C})|}$ having a $1$ in the coordinate indexed by $a$ and a $0$ elsewhere.If $C = [a,b] \in \mathcal{P}$ is a cell with diagonal corners $a,b$ and anti-diagonal corners $c,d$, we set $$\mathbf{v}_{[a,b]} = \mathbf{v}_a + \mathbf{v}_b - \mathbf{v}_c - \mathbf{v}_d \in \mathbb{Z}^{|V(\mathcal{C})|}.$$ We define $\Lambda_{\mathcal{P}}$ as the sublattice of $\mathbb{Z}^{|V(\mathcal{P})|}$ generated by the vectors $\mathbf{v}_C$ for all $C \in \mathcal{P}$. Observe that the rank of $\Lambda_{\mathcal{P}}$ equals $|\mathcal{P}|$.
Let $n = |V(\mathcal{P})|$. For $\mathbf{v} \in \mathbb{N}^n$, denote by $x^{\mathbf{v}}$ the monomial in $S$ having $\mathbf{v}$ as its exponent vector. For $\mathbf{e} \in \mathbb{Z}^n$, let $\mathbf{e}^+$ be the vector obtained from $\mathbf{e}$ by replacing all negative components with zero, and set $\mathbf{e}^- = - (\mathbf{e} - \mathbf{e}^+)$ for its non-positive part.
The lattice ideal $L_{\mathcal{P}}$ of $\Lambda_{\mathcal{P}}$ is then the binomial ideal in $S$ defined as$$L_{\mathcal{P}} = (\,x^{\mathbf{e}^+} - x^{\mathbf{e}^-} \mid \mathbf{e} \in \Lambda_{\mathcal{P}}\,).$$

i1 : Q = cellCollection {{1,1},{2,1},{3,1},{3,2},{1,2},{1,3},{2,3}};

i2 : L = polyoLattice Q o2 = ideal (- x x + x x , - x x + x x , - x x + 1,1 4,2 1,2 4,1 2,1 4,2 2,2 4,1 3,1 4,2 ------------------------------------------------------------------------ x x , - x x + x x , - x x + x x , - x x + 3,2 4,1 2,3 1,4 2,4 1,3 3,3 1,4 3,4 1,3 2,2 1,4 ------------------------------------------------------------------------ x x , - x x + x x , - x x + x x , - x x + 2,4 1,2 2,2 1,3 2,3 1,2 2,1 1,4 2,4 1,1 2,1 1,3 ------------------------------------------------------------------------ x x , - x x + x x , - x x + x x , - x x + 2,3 1,1 2,1 1,2 2,2 1,1 3,1 1,2 3,2 1,1 3,3 2,4 ------------------------------------------------------------------------ x x , - x x + x x , x x - x x , x x - 3,4 2,3 3,1 2,2 3,2 2,1 3,2 4,3 3,3 4,2 3,1 4,3 ------------------------------------------------------------------------ x x ) 3,3 4,1 o2 : Ideal of QQ[x ..x , x ..x , x ..x , x ..x ] 3,1 3,4 2,1 2,4 1,1 1,4 4,1 4,3

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