Let $\mathcal{L}$ be a finite distributive lattice and $K[\mathcal{L}]=K[x_\alpha: \alpha \in \mathcal{L}]$ be the polynomial ring over $K$ and $I_{\mathcal{L}}=\langle x_\alpha x_\beta- x_{\alpha\vee \beta} x_{\alpha\wedge\beta} : \alpha \nsim \beta,\alpha,\beta \in {\mathcal{L}} \rangle$ be the ideal of $K[\mathcal{L}]$. In this article we describe the first syzygy of Hibi ring $R[\mathcal{L}]=K[\mathcal{L}]/I_{\mathcal{L}}$, for a distributive lattice. We also derive an exact formula for the first betti number of a planar distributive lattice. 查看全文>>