Möbius function of semigroup posets through Hilbert series

In this paper, we investigate the Mobius function µ S associated to a (locally finite) poset arising from a semigroup S of Z m . We introduce and develop a new approach to study µ S by using the Hilbert series of S . The latter enables us to provide formulas for µ S when S belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.

[1]  B. Sturmfels,et al.  Binomial Ideals , 1994, alg-geom/9401001.

[2]  Isabel Bermejo,et al.  Complete intersections in simplicial toric varieties , 2015, J. Symb. Comput..

[3]  James A. Deddens.  A Combinatorial Identity Involving Relatively Prime Integers , 1979, J. Comb. Theory, Ser. A.

[5]  I. Sengupta,et al.  Criterion for Complete Intersection of certain Monomial Curves , 2003 .

[6]  Martin Kreuzer,et al.  Computational Commutative Algebra 1 , 2000 .

[7]  D. Knuth,et al.  Mathematics for the Analysis of Algorithms , 1999 .

[8]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[9]  Apostolos Thoma,et al.  Binomial generation of the radical of a lattice ideal , 2008, 0811.3833.

[10]  Walter D. Morris,et al.  Affine semigroup rings that are complete intersections , 1997 .

[11]  Jürgen Herzog,et al.  Generators and relations of abelian semigroups and semigroup rings , 1970 .

[12]  Complete intersections in certain affine and projective monomial curves , 2014, 1407.7007.

[13]  G. Rota.  On the foundations of combinatorial theory I. Theory of Möbius Functions , 1964 .

[14]  G. Rota.  On the Foundations of Combinatorial Theory , 2009 .

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  Affine semigroups having a unique Betti element , 2012, 1203.4138.

[17]  Jorge Luis Ramírez Alfonsín,et al.  On the Möbius function of the locally finite poset associated with a numerical semigroup , 2012, 1209.1880.

[18]  L. O'Carroll.  GRÖBNER BASES AND CONVEX POLYTOPES (University Lecture Series 8) , 1997 .

[19]  Ramírez Alfonsin,et al.  The diophantine frobenius problem , 2005 .

[20]  Anargyros Katsabekis,et al.  AN INDISPENSABLE CLASSIFICATION OF MONOMIAL CURVES IN A 4 .k , 2014 .

[21]  C. Marijuán,et al.  Minimal systems of generators for ideals of semigroups , 1998 .

[22]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .