Searcharxiv⌕ Search

arXiv subjects

Rahim Rahmati-Asghar

Publications and source records attributed to Rahim Rahmati-Asghar.

9 recordsLinked to original sources
arXiv2018

On the polymatroidal property of monomial ideals with a view towards orderings of minimal generators

We prove that a monomial ideal $I$ generated in a single degree, is polymatroidal if and only if it has linear quotients with respect to the lexicographical ordering of the minimal generators induced by every ordering of variables. We also conjecture that the polymatroidal ideals can be characterized with linear quotients property with respect to the reverse lexicographical ordering of the minimal generators induced by every ordering of variables. We prove our conjecture in many special cases.

math.AC↗
arXiv2017

Expansion and contraction functors on matriods

Let $M$ be a matroid. We study the expansions of $M$ mainly to see how the combinatorial properties of $M$ and its expansions are related to each other. It is shown that $M$ is a graphic, binary or a transversal matroid if and only if an arbitrary expansion of $M$ has the same property. Then we introduce a new functor, called contraction, which acts in contrast to expansion functor. As a main result of paper, we prove that a matroid $M$ satisfies White's conjecture if and only if an arbitrary expansion of $M$ does. It follows that it suffices to focus on the contraction of a given matroid for checking whether the matroid satisfies White's conjecture. Finally, some classes of matroids satisfying White's conjecture are presented.

math.CO↗
arXiv2017

Pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes

We introduce pretty $k$-clean monomial ideals and $k$-decomposable multicomplexes, respectively, as the extensions of the notions of $k$-clean monomial ideals and $k$-decomposable simplicial complexes. We show that a multicomplex $Γ$ is $k$-decomposable if and only if its associated monomial ideal $I(Γ)$ is pretty $k$-clean. Also, we prove that an arbitrary monomial ideal $I$ is pretty $k$-clean if and only if its polarization $I^p$ is $k$-clean. Our results extend and generalize some results due to Herzog-Popescu, Soleyman Jahan and the current author.

math.AC↗
arXiv2017

$k$-clean monomial ideals

In this paper, we introduce the concept of $k$-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of $k$-clean ideals, we show that a $(d-1)$-dimensional simplicial complex is $k$-decomposable if and only if its Stanley-Reisner ideal is $k$-clean, where $k\leq d-1$. We prove that the classes of monomial ideals like monomial complete intersection ideals, Cohen-Macaulay monomial ideals of codimension 2 and symbolic powers of Stanley-Reisner ideals of matroid complexes are $k$-clean for all $k\geq 0$.

math.AC↗
arXiv2017

On the facet ideal of an expanded simplicial complex

For a simplicial complex $Δ$, the affect of the expansion functor on combinatorial properties of $Δ$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal $I(Δ)$ and its Alexander dual which we denote by $J_Δ$ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion $Δ^α$ the ideals $J_Δ$ and $J_{Δ^α}$ have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals $I(Δ)$ and $I(Δ^α)$ are equal. Moreover, the projective dimensions of $I(Δ)$ and $I(Δ^α)$ are compared. In the sequel for a graph $G$, some properties that are equivalent in $G$ and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph $G$, we give some conditions for adding or removing a vertex from $G$, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).

math.AC↗
arXiv2017

$k$-shellable simplicial complexes and graphs

In this paper we show that a $k$-shellable simplicial complex is the expansion of a shellable complex. We prove that the face ring of a pure $k$-shellable simplicial complex satisfies the Stanley conjecture. In this way, by applying expansion functor to the face ring of a given pure shellable complex, we construct a large class of rings satisfying the Stanley conjecture. Also, by presenting some characterizations of $k$-shellable graphs, we extend some results due to Castrillón-Cruz, Cruz-Estrada and Van Tuyl-Villareal.

math.AC↗
arXiv2016

On the Stanley-Reisner ideal of an expanded simplicial complex

Let $Δ$ be a simplicial complex. We study the expansions of $Δ$ mainly to see how the algebraic and combinatorial properties of $Δ$ and its expansions are related to each other. It is shown that $Δ$ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or $k$-decomposable, if and only if an arbitrary expansion of $Δ$ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of $Δ$ and those of their expansions are compared.

math.AC↗
arXiv2015

The behaviors of expansion functor on monomial ideals and toric rings

In this paper we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness and having linear quotients are preserved under taking the expansion functor. The main part of the paper is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroid $P$, if toric ideal of $P$ is generated by double swaps then toric ideal of any expansion of $P$ has such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated.

math.AC↗