Discrete piecewise linear functions

The concept of permutograph is introduced and properties of integral functions on permutographs are investigated. The central result characterizes the class of integral functions that are representable as lattice polynomials. This result is used to establish lattice polynomial representations of piecewise linear functions on R^d and continuous selectors on linear orders.

[1]  R. Stanley.  An Introduction to Hyperplane Arrangements , 2007 .

[2]  P. Cameron,et al.  PERMUTATION GROUPS , 2019, Group Theory for Physicists.

[3]  S. Ovchinnikov,et al.  Integral Representation of Invariant Functionals , 2000 .

[4]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[5]  James Stuart Tanton,et al.  Encyclopedia of Mathematics , 2005 .

[6]  J. M. Tarela,et al.  Region configurations for realizability of lattice Piecewise-Linear models , 1999 .

[7]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[8]  Sergei Ovchinnikov,et al.  Hyperplane arrangements in preference modeling , 2005 .

[9]  Bernd Sturmfels,et al.  Oriented Matroids: Notation , 1999 .

[10]  Peter L. Hammer,et al.  Boolean Functions - Theory, Algorithms, and Applications , 2011, Encyclopedia of mathematics and its applications.

[11]  Sven G. Bartels,et al.  Continuous selections of linear functions and nonsmooth critical point theory , 1995 .

[12]  Claude Berge.  Principles of Combinatorics , 2012 .

[13]  Sergei Ovchinnikov.  Boolean Representation of Manifolds and Functions , 2001 .

[14]  Shuning Wang,et al.  A compact f-f model of high-dimensional piecewise-linear function over a degenerate intersection , 2005, IEEE Transactions on Circuits and Systems I: Regular Papers.

[15]  S. Ovchinnikov.  Means on ordered sets , 1996 .

[16]  G. Birkhoff,et al.  Lattice-ordered Rings , 1940 .

[17]  Shuning Wang,et al.  General constructive representations for continuous piecewise-linear functions , 2004, IEEE Trans. Circuits Syst. I Regul. Pap..

[18]  G. Ziegler.  Lectures on Polytopes , 1994 .

[19]  Sergei Ovchinnikov,et al.  Invariant Functions on Simple Orders , 1997 .

[20]  Louis Mahe.  On the Pierce-Birkhoff conjecture , 1984 .

[21]  P. Rosenstiehl,et al.  Analyse algébrique d'un scrutin , 1963 .

[22]  Sergei Ovchinnikov,et al.  3 S ep 2 00 0 Max – Min Representation of Piecewise Linear Functions , 2008 .

[23]  D. Melzer,et al.  On the expressibility of piecewise-linear continuous functions as the difference of two piecewise-linear convex functions , 1986 .

[24]  George Gratzer,et al.  Universal Algebra , 1979 .

[25]  Robert H. Wilkinson,et al.  A Method of Generating Functions of Several Variables Using Analog Diode Logic , 1963, IEEE Trans. Electron. Comput..

[26]  B. Sturmfels.  Oriented Matroids , 1993 .

[27]  A. M. W. Glass,et al.  Ordered Permutation Groups , 1982 .

[28]  J. Isbell,et al.  Lattice-ordered rings and function rings. , 1962 .

[29]  Mark Spivakovsky,et al.  A CONNECTEDNESS THEOREM FOR REAL SPECTRA OF POLYNOMIAL RINGS , 2006 .

[30]  N. Biggs.  Algebraic Graph Theory , 1974 .