Edgewise Cohen-Macaulay connectivity of partially ordered sets


  • Christos A. Athanasiadis
  • Myrto Kallipoliti




The proper parts of face lattices of convex polytopes are shown to satisfy a strong form of the Cohen-Macaulay property, namely that removing from their Hasse diagram all edges in any closed interval results in a Cohen-Macaulay poset of the same rank. A corresponding notion of edgewise Cohen-Macaulay connectivity for partially ordered sets is investigated. Examples and open questions are discussed.


Adiprasito, K. A. and Björner, A., Filtered geometric lattices and Lefschetz section theorems over the tropical semiring, preprint arXiv:1401.7301, 2014.

Athanasiadis, C. A., Brady, T., and Watt, C., Shellability of noncrossing partition lattices, Proc. Amer. Math. Soc. 135 (2007), no. 4, 939–949. https://doi.org/10.1090/S0002-9939-06-08534-0

Baclawski, K., Cohen-Macaulay connectivity and geometric lattices, European J. Combin. 3 (1982), no. 4, 293–305. https://doi.org/10.1016/S0195-6698(82)80014-0

Björner, A., Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159–183. https://doi.org/10.2307/1999881

Björner, A., Topological methods, in “Handbook of combinatorics”, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819--1872.

Björner, A., “Let ∆ be a Cohen-Macaulay complex $dots $”, in “The mathematical legacy of Richard P. Stanley”, Amer. Math. Soc., Providence, RI, 2016, pp. 105--118.

Bruggesser, H. and Mani, P., Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197–205. https://doi.org/10.7146/math.scand.a-11045

Kallipoliti, M. and Kubitzke, M., A poset fiber theorem for doubly Cohen-Macaulay posets and its applications, Ann. Comb. 17 (2013), no. 4, 711–731. https://doi.org/10.1007/s00026-013-0203-8

Matsuoka, N. and Murai, S., Uniformly Cohen-Macaulay simplicial complexes and almost Gorenstein$^*$ simplicial complexes, J. Algebra 455 (2016), 14–31. https://doi.org/10.1016/j.jalgebra.2016.02.005

Munkres, J. R., Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984.

Stanley, R. P., Combinatorics and commutative algebra, second ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996.

Stanley, R. P., Enumerative combinatorics. Vol. 1, second ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012.

Wachs, M. L., Poset topology: tools and applications, in “Geometric combinatorics”, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 497--615.

Wachs, M. L. and Walker, J. W., On geometric semilattices, Order 2 (1986), no. 4, 367–385. https://doi.org/10.1007/BF00367425

Walker, J. W., Topology and combinatorics of ordered sets, Ph.D. thesis, Massachussetts Institute of Technology, 1981.

Welker, V., On the Cohen-Macaulay connectivity of supersolvable lattices and the homotopy type of posets, European J. Combin. 16 (1995), no. 4, 415–426. https://doi.org/10.1016/0195-6698(95)90021-7




How to Cite

Athanasiadis, C. A., & Kallipoliti, M. (2018). Edgewise Cohen-Macaulay connectivity of partially ordered sets. MATHEMATICA SCANDINAVICA, 122(1), 5–17. https://doi.org/10.7146/math.scand.a-97270