Shahrood University of Technology Journal of Algebraic Systems 2345-5128 14 3 2026 06 01 VERTEX DECOMPOSABILITY AND WEAKLY POLYMATROIDAL IDEALS 427 438 3820 10.22044/jas.2024.14707.1854 EN Amir Mafi Department of Mathematics, University of Kurdistan, P.O. Box 416, Sanandaj, Iran. Dler Naderi Department of Mathematics, University of Kurdistan, P.O. Box 416, Sanandaj, Iran. HeroHero Saremi Department of Mathematics, Sa.C., Islamic Azad University, Sanandaj, Iran. Journal Article 2024 06 23 ‎Let $K$ be a field and $R=K[x_1,\ldots‎, ‎x_n]$ 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.<br />‎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 $u_1,\ldots,u_s$ such that $s\leq 3$ or $\supp(u_i)\cup\supp(u_j)=\{x_1,\ldots,x_n\}$ for all $i\neq j$‎, ‎then $\Delta$ is vertex decomposable‎. ‎Furthermore‎, ‎we prove that‎<br />‎if $I$ is a monomial ideal of degree $2$ then $I$ is weakly polymatroidal if and only if $I$ has linear quotients if and only if $I$ is vertex splittable‎. https://jas.shahroodut.ac.ir/article_3820_b6c502776ab04f365619fe0d0cb3ed57.pdf