Generalization of Shih-Dong's combinational fixed point theorem to finite distributive lattices (Q2789185)

\textit{M.-H. Shih} and \textit{J.-L. Dong} [Adv. Appl. Math. 34, No. 1, 30--46 (2005; Zbl 1060.05505)] have proved that if a map from \(\{0,1\}^n\) to itself has the property that all the boolean eigenvalues of the discrete Jacobian matrix of each element of \(\{0,1\}^n\) are zero, then it has a unique fixed point. This theorem answers the ``Combinatorial Fixed Point Conjecture'', a combinatorial version of the Jacobian conjecture. In the present paper, the authors provide an extension of Shih-Dong's theorem to all finite distributive lattices.NEWLINENEWLINEThe authors prove the following theorem: Let \(L\) be a finite distributive lattice with \(\mathcal{J}(L)=\{a^1,\dots,a^n\}\) and \(F:L\rightarrow L\) be such that \(\Gamma\left(F'(x)\right)\) has no circuit for all \(x\in L\). Then for every \(a,b\in L\) with \(a<b\), \(F\) has a unique local fixed point \(\alpha\in [a,b]\). Also, they prove the following theorem: Let \(L\) be a finite distributive lattice and \(F:L\rightarrow L\). Let \(D\) be a cyclic attractor of \(G(F)\). Then \(\bigcup_{x\in L}\Gamma\left(F'(x)\right)\) contains a negative circuit.