Abstract
Motivated by the classical Frobenius problem, we introduce the Frobenius poset on the integers \({\mathbb Z}\) , that is, for a sub-semigroup Λ of the non-negative integers (\({\mathbb N}\) , +), we define the order by n ≤Λ m if \({{m-n \in \Lambda}}\). When Λ is generated by two relatively prime integers a and b, we show that the order complex of an interval in the Frobenius poset is either contractible or homotopy equivalent to a sphere. We also show that when Λ is generated by the integers {a, a + d, a + 2d, . . . , a + (a−1)d}, the order complex is homotopy equivalent to a wedge of spheres.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barvinok A.: A Course in Convexity. American Mathematical Society, Providence, RI (2002)
Beck M., Robins S.: Computing the Continuous Discretely. Springer, New York (2007)
Billera L.J., Hetyei G.: Decompositions of partially ordered sets. Order 17(2), 141–166 (2000)
Billera L.J., Myers A.N.: Shellability of interval orders. Order 15(2), 113–117 (1999)
Björner A., Welker V.: Segre and Rees products of posets, with ring-theoretic applications. J. Pure Appl. Algebra 198(1-3), 43–55 (2005)
Collins K.L.: Planar lattices are lexicographically shellable. Order 8(4), 375–381 (1992)
Ehrenborg R., Hetyei G.: The topology of the independence complex. European J. Combin. 27(6), 906–923 (2006)
Forman R.: Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998)
Forman, R.: A user’s guide to discrete Morse theory. Sém. Lothar. Combin. 48, Art. B48c (2002)
Hersh, P., Welker, V.: Gröbner basis degree bounds on \({{\rm Tor}^{k [ \Lambda ]}_{\bullet}(k, k){_\bullet}}\) and discrete Morse theory for posets. In: Barvinok, A. et al (eds.) Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization. Contemp. Math. Vol. 374, pp. 101–138. Amer. Math. Soc., Providence, RI (2005)
Kozlov D.N.: Complexes of directed trees. J. Combin. Theory Ser. A 88(1), 112–122 (1999)
Kozlov D.N.: Combinatorial Algebraic Topology. Springer, Berlin (2008)
Kozlov D.N.: Discrete Morse theory and Hopf bundles. Pacific J. Math. 249(2), 371–376 (2011)
Laudal O.A., Sletsjøe A.: Betti numbers of monoid algebras. Applications to 2-dimensional torus embeddings. Math. Scand. 56, 145–162 (1985)
Ong, D., Ponomarenko, V.: The Frobenius number of geometric sequences. Integers 8, #A33 (2008)
Peeva I., Reiner V., Sturmfels B.: How to shell a monoid. Math. Ann. 310(2), 379–393 (1998)
Roberts J.B.: Note on linear forms. Proc. Amer. Math Soc. 7, 465–469 (1956)
Stamate, D.I.: Computational algebra and combinatorics in commutative algebra. PhD Thesis, University of Bucharest (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Clark, E., Ehrenborg, R. The Frobenius Complex. Ann. Comb. 16, 215–232 (2012). https://doi.org/10.1007/s00026-012-0127-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00026-012-0127-8