arXiv subjects
Publications and source records attributed to Tongsuo Wu.
Let $\D$ be a $(d-1)$-dimensional pure $f$-simplicial complex over vertex set $[n]$. In this paper, it is proved that $n=2d$ holds true if $\D$ is minimal Cohen-Macaulay. It is also indicated that the recent work of \cite{Dao2020} implies that shellable condition on a pure simplicial complex $\D$ is identical with CM properties of a full series of subcomplexes of $\D$.
For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where $[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $Δ$ such that ${\rm dim}(Δ)={\rm dim}(Δ^c)$ and $\mathcal{F}(Δ)\cap \mathcal{F}(Δ^c)=\emptyset$, we prove that the facet ideal $I(Δ)$ is Cohen-Macaulay if and only if it has linear resolution. For a $d$-dimensional pure $f$-simplicial complex $Δ$ such that $Δ'=:\langle F\mid F\in [n]_d\smallsetminus \mathcal F(Δ)\rangle$ is an $f$-simplicial complex, we prove that $I(Δ^c)$ is Cohen-Macaulay if and only if $I(Δ')$ has linear resolution.
Let $R$ be a standard graded algebra over an infinite field $\mathbb{K}$ and $M$ a finitely generated $\ZZ$-graded $R$-module. Let $I_1,\ldots I_m$ be graded ideals of $R$. The functions $r(M/I_1^{a_1}\ldots I_m^{a_m}M)$ and $r(I_1^{a_1}\ldots I_m^{a_m}M)$ are investigated and their asymptotical behaviours are given. Here $r(\bullet)$ stands for the reduction number of a finitely graded module $\bullet$.
In this note, we use the concept of a polynomial ring to give an elementary proof to Cayley-Hamilton Theorem. We also give an elementary proof to Birkhoff theorem on Bi-stochastic matrices.
For every simple graph $G$, a class of multiple clique cluster-whiskered graphs $G^{md}$ is introduced, and it is shown that all graphs $G^{md}$ are vertex decomposable, thus the independence simplicial complex ${\rm Ind}\,G^{md}$ is sequentially Cohen-Macaulay; the properties of the graphs $G^{md}$ and the clique-whiskered graph $G^π$ are studied, including the enumeration of facets of the complex ${\rm Ind}\, G^π$ and, the calculation of Betti numbers of the cover ideal $I_c(G^{md})$.
For each Boolean graph $B_n$, it is proved that both $B_n$ and its complement graph $\overline{B_n}$ are vertex decomposable. It is also proved that $B_n$ is an unmixed graph, thus it is also Cohen-Macaulay.
Let $I$ be a graded ideal of $K[x_1,\ldots,x_n]$ generated by homogeneous polynomials of a same degree $d$, and assume that $I$ has linear quotients. In this note, we use Horseshoe Lemma to give a relatively direct inductive construction of a minimal free resolution of $I$, which is called a $d$-linear resolution.
Imposing a strong condition on the linear order of shellable complexes, we introduce strong shellability. Basic properties, including the existence of dimension-decreasing strong shelling orders, are developed with respect to nonpure strongly shellable complexes. Meanwhile, pure strongly shellable complexes can be characterized by the corresponding codimension one graphs. In addition, we show that the facet ideals of pure strongly shellable complexes have linear quotients.
When $\mathcal{C}$ is a chordal clutter in the sense of Woodroofe or Emtander, we show that the complement clutter is edgewise strongly shellable. When $\mathcal{C}$ is indeed a finite simple graph, we study various characterizations of chordal graphs from the point of view of strong shellability. In particular, the generic graph $G_T$ of a tree is shown to be bi-strongly shellable. We also characterize edgewise strongly shellable bipartite graphs in terms of constructions from upward sequences. \end{abstract}
A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A po-semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a po-semiring are studied. In particular, it is proved that under some mild assumption the set $Z(A)$ of nonzero zero divisors of $A$ is $A\setminus \{0,1\}$, each prime element of $A$ is a maximal element, and the zero divisor graph $\G(A)$ of $A$ is a finite graph if and only if $A$ is finite. For a po-semiring $A$ with $Z(A)=A\setminus \{0,1\}$, it is proved that $A$ has finitely many maximal elements if ACC holds either for elements of $A$ or for principal annihilating ideals of $A$. As applications of prime elements, it is shown that the structure of a po-semiring $A$ is completely determined by the structure of integral po-semirings if either $|Z(A)|=1$ or $|Z(A)|=2$ and $Z(A)^2\not=0$. Applications to the ideal structure of commutative rings are considered.
For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik ideal. In this paper, we characterize the Lyubeznik ideals, and we discover some classes of Lyubeznik ideals.
A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $δ_{\mathcal{F}}(I)$ and $δ_{\mathcal{N}}(I)$ have the same $f$-vector, where $δ_{\mathcal{F}}(I)$ ($δ_{\mathcal{N}}(I)$, respectively) is the facet (Stanley-Reisner, respectively) complex related to $I$. In this paper, we introduce and study perfect subsets of $2^{[n]}$ and use them to characterize the $f$-ideals of degree $d$. We give a decomposition of $V(n, 2)$ by taking advantage of a correspondence between graphs and sets of square-free monomials of degree $2$, and then give a formula for counting the number of $f$-ideals of degree $2$, where $V(n, 2)$ is the set of $f$-ideals of degree 2 in $K[x_1,\ldots,x_n]$. We also consider the relation between an $f$-ideal and an unmixed monomial ideal.
A square-free monomial ideal $I$ is called an {\it $f$-ideal}, if both $δ_{\mathcal{F}}(I)$ and $δ_{\mathcal{N}}(I)$ have the same $f$-vector, where $δ_{\mathcal{F}}(I)$ ($δ_{\mathcal{N}}(I)$, respectively) is the facet (Stanley-Reisner, respectively) complex related to $I$. In this paper, we introduce the concepts of perfect set containing $k$ and perfect set without $k$. We study the $(n, d)^{th}$ perfect sets and show that $V(n, d) \neq \emptyset$ for $d \geq 2$ and $n \geq d+2$. Then we give some algorithms to construct $(n, d)^{th}$ $f$-ideals and show an upper bound for the $(n, d)^{th}$ perfect number.
In this paper, we show for a monomial ideal $I$ of $K[x_1,x_2,\ldots,x_n]$ that the integral closure $\ol{I}$ is a monomial ideal of Borel type (Borel-fixed, strongly stable, lexsegment, or universal lexsegment respectively), if $I$ has the same property. We also show that the $k^{th}$ symbolic power $I^{(k)}$ of $I$ preserves the properties of Borel type, Borel-fixed and strongly stable, and $I^{(k)}$ is lexsegment if $I$ is stably lexsegment. For a monomial ideal $I$ and a monomial prime ideal $P$, a new ideal $J(I, P)$ is studied, which also gives a clear description of the primary decomposition of $I^{(k)}$. Then a new simplicial complex $_J\bigtriangleup$ of a monomial ideal $J$ is defined, and it is shown that $I_{_J\bigtriangleup^{\vee}} = \sqrt{J}$. Finally, we show under an additional weak assumption that a monomial ideal is universal lexsegment if and only if its polarization is a squarefree strongly stable ideal.
Let $G=Γ(S)$ be a semigroup graph, i.e., a zero-divisor graph of a semigroup $S$ with zero element 0. For any adjacent vertices $x, y$ in $G$, denote $C(x,y)={z\in V(G) | N(z)={x,y}}$. Assume that in $G$ there exist two adjacent vertices $x,y$, a vertex $s\in C(x,y)$ and a vertex $z$ such that $d(s,z)=3$. In this paper, we study algebraic properties of $S$ with such graphs $G=\G(S)$, giving some sub-semigroups and ideals of $S$. We construct some classes of such semigroup graphs and classify all semigroup graphs with the property in two cases.
A ring $R$ is called a PIR, if each ideal of $R$ is a principal ideal. An local ring $(R,\mf{m)}$ is a artinian PIR if and only if its maximal ideal $\mf{m}$ is principal and has finite nilpotency index. In this paper, we determine the structure of a finite local PIR.
In this paper, we determine bipartite graphs and complete graphs with horns, which are realizable as zero-divisor graphs of po-semirings. As applications, we classify commutative rings $R$ whose annihilating-ideal graph $\mathbb {AG}(R)$ are either bipartite graphs or complete graphs with horns.
This paper studies the co-maximal graph $\Om(R)$, the induced subgraph $\G(R)$ of $\Om(R)$ whose vertex set is $R\setminus (U(R)\cup J(R))$ and a retract $\G_r(R)$ of $\G(R)$, where $R$ is a commutative ring. We show that the core of $\G(R)$ is a union of triangles and rectangles, while a vertex in $\G(R)$ is either an end vertex or a vertex in the core. For a non-local ring $R$, we prove that both the chromatic number and clique number of $\G(R)$ are identical with the number of maximal ideals of $R$. A graph $\G_r(R)$ is also introduced on the vertex set $\{Rx|\,x\in R\setminus (U(R)\cup J(R))\}$, and graph properties of $\G_r(R)$ are studied.