施-董的組合不動點定理證明，如果從n維超立方體到自身的函數滿足了每個在超立方體的元素其布爾雅可比矩陣的特徵值是零，那麼該函數有唯一的固定點。該定理等價對偶敘述具有生物學意義。我們的目標是推廣施-董定理到所有的有限分配格。我們的證明方法是基於施-董的“集體影響法”以及G.伯克霍夫的有限分配格表現定理。

Shih-Dong's combinational fixed point theorem asserts that if a map from the n-dimensional hypercube into itself satisfies that all the Boolean eigenvalues of the Boolean Jacobian matrix are zero for each element in the hypercube, then it has a unique fixed point. Its equivalent contrapositive form has biological implications. Our goal is to provide an extension of Shih-Dong's theorem into all finite distributive lattices. Our method of proof is based on Shih-Dong's “collective effect method” as well as G. Birkhoff's representation theorem for finite distributive lattices.

[1] H. Bass, E. Connell, and D.Wright, The Jacobian conjecture: reduction on degree and formal expansion of the inverse., Bull. Amer. Math. Soc., 7 No.2 (1982), 287–330.
[2] G. Birkhoff, Rings of sets, Duke Mathematical Journal, 3 (1937), 443–454.
[3] G. Birkhoff, Lattice Theory, Third ed., Amer. Math. Soc., Providence, Rhode Island, 1967.
[4] A. Cima, A. Gasull, and F. Ma˜nosas, The discrete Markus-Yamabe problem, Nonlinear analysis, 35 (1999), 343–354.
[5] O. H. Keller, Ganze Cremona-Transformationen, Monatshefte f¨ur Mathematik und Physik, 47 Issue 1 (1939), 299–306.
[6] E. Remy, P. Ruet, and D. Thieffry, Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework, Advances in Applied mathematics, 41 (2008), 335–350.
[7] A. Richard and J.-P. Comet, Necessary conditions for multistationarity in discrete dynamical systems, Discrete Applied Mathematics, 155 (2007), 2403–2413.
[8] A. Richard, An extension of a combinatorial fixed point theorem of Shih and Dong, Advances in Applied mathematics, 41 (2008), 620–627.
[9] A. Richard, Positive circuits and maximal number of fixed points in discrete dynamical systems, Discrete Applied Mathematics, 157 (2009), 3281–3288.
[10] F. Robert, Discrete Iteration, A Metric Study, Springer Ser. Comput. Math., vol. 6, Springer-Verlag, Berlin-Heidelberg-New York, 1986.
[11] W. Rudin, Injective polynomial maps are automorphisms, Amer. Math. Monthly, 102 No.6 (1995), 540–543.
[12] M.-H. Shih and J.-L. Ho, Solution of the Boolean Markus-Yamabe problem, Advances in Applied Mathematics, 22 (1999), 60–102.
[13] M.-H. Shih and J.-L. Dong, A combinatorial analogue of the Jacobian problem in automata networks, Advances in Applied Mathematics, 34 (2005), 30–46.
[14] S. Smale, Mathematical problems for the next century, Mathematical Intelligencer, 20 No. 2 (1998), 7–15.
[15] C. Soul´e, Mathematical approaches to differentiation and gene regulation, C.R. Paris Biologies, 329 (2006), 13–20.
[16] R. Thomas, On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations, Springer Series in Synergetics, 9 (1981), 180–193.
[17] R. Thomas and R. d’Ari, Biological Feedback, CRC Press, 1990.
[18] R. Thomas, Laws for the dynamics of regulatory networks, Internat. J. Dev. Biol., 42 (1998), 479–485.
[19] R. Thomas and M. Kaufman, Multistationarity, the basis of cell differentiation and memory. I. and II., Chaos, 11 (2001) 170-195.