VERTEX DECOMPOSABILITY AND WEAKLY POLYMATROIDAL IDEALS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, University of Kurdistan, P.O. Box 416, Sanandaj, Iran.

2 Department of Mathematics, Sa.C., Islamic Azad University, Sanandaj, Iran.

10.22044/jas.2024.14707.1854

Abstract

‎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.
‎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
‎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.

Keywords

References

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Journal of Algebraic Systems
Volume 14, Issue 3
June 2026
Pages 427-438

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