We study chains of nonzero edge ideals that are invariant under the action of the monoid \({{\,\textrm{Inc}\,}}\) of increasing functions on the positive integers. We prove that the sequence of Castelnuovo–Mumford regularity of ideals in such a chain is eventually constant with limit either 2 or 3, and we determine explicitly when the constancy behavior sets in. This provides further evidence to a conjecture on the asymptotic linearity of the regularity of \({{\,\textrm{Inc}\,}}\)-invariant chains of homogeneous ideals. The proofs reveal unexpected combinatorial properties of \({{\,\textrm{Inc}\,}}\)-invariant chains of edge ideals.

我们研究在正整数上递增函数的幺半群\({{\,\textrm{Inc}\,}}\)作用下不变的非零边缘理想链。我们证明了这样一条链中理想的 Castelnuovo-Mumford 正则序列最终在极限为 2 或 3 时是恒定的，并且我们明确地确定了恒定行为何时开始。这为关于链的渐近线性的猜想提供了进一步的证据。\({{\,\textrm{Inc}\,}}\)的正则性- 齐次理想的不变链。证明揭示了\({{\,\textrm{Inc}\,}}\)边缘理想不变链的意外组合属性。