VERTEX DECOMPOSABILITY AND WEAKLY POLYMATROIDAL IDEALS

‎Let K be a field and R=K[x_۱,\ldots‎, ‎x_n] be the polynomial ring in n variables over a field K‎. ‎Let \Delta be a simplicial complex on n vertices and I=I_{\Delta} be its Stanley-Reisner ideal‎.‎In this paper‎, ‎we show that if I is a matroidal ideal then the following conditions are equivalent‎: ‎(i) \Delta is sequentially Cohen-Macaulay; (ii) \Delta is shellable; (iii) \Delta is vertex decomposable‎. ‎Also‎, ‎if I is a minimally generated by u_۱,\ldots,u_s such that s\leq ۳ or \supp(u_i)\cup\supp(u_j)=\{x_۱,\ldots,x_n\} for all i\neq j‎, ‎then \Delta is vertex decomposable‎. ‎Furthermore‎, ‎we prove that‎‎if I is a monomial ideal of degree ۲ then I is weakly polymatroidal if and only if I has linear quotients if and only if I is vertex splittable‎.

کلیدواژه ها:

Vertex decomposable‎ ، Shellable ، ‎weakly polymatroidal