This abstract is essentially taken from the introduction of the paper Monomial and toric ideals associated to Ferrers graphs [13], written jointly with Alberto Corso. A Ferrers graph is a bipartite graph on two distinct vertex sets X = {x1, . . . , xn} and Y = {y1, . . . , ym} such that if (xi, yj) is an edge of G, then so is (xp, yq) for 1 ≤ p ≤ i and 1 ≤ q ≤ j. In addition, (x1, ym) and (xn, ...

An old question of T. Jech and K. Prikry asks if an existence of a precipitous ideal implies necessary existence of a normal precipitous ideal. The aim of the paper is to prove some results in the positive direction. Thus, it is shown that under some mild assumptions, an existence of a precipitous ideal over א1 implies an existence of a normal precipitous ideal over א1 once a Cohen subset is ad...

Let G be a simple graph on the vertex set [n] with edge set E(G) and let S be the polynomial ring K[x1, . . . , xn, y1, . . . , yn] in 2n variables endowed with the lexicographic order induced by x1 > · · · > xn > y1 > · · · > yn. The binomial edge ideal JG ⊂ S associated with G is generated by all the binomials fij = xiyj−xjyi with {i, j} ∈ E(G). The binomial edge ideals were introduced in [5]...

the rings considered in this article are commutative with identity $1neq 0$. by a proper ideal of a ring $r$,  we mean an ideal $i$ of $r$ such that $ineq r$.  we say that a proper ideal $i$ of a ring $r$ is a  maximal non-prime ideal if $i$ is not a prime ideal of $r$ but any proper ideal $a$ of $r$ with $ isubseteq a$ and $ineq a$ is a prime ideal. that is, among all the proper ideals of $r$,...

in this article, we give several generalizations of the concept of annihilating ideal graph over a commutative ring with identity to modules. weobserve that over a commutative ring $r$, $bbb{ag}_*(_rm)$ isconnected and diam$bbb{ag}_*(_rm)leq 3$. moreover, if $bbb{ag}_*(_rm)$ contains a cycle, then $mbox{gr}bbb{ag}_*(_rm)leq 4$. also for an $r$-module $m$ with$bbb{a}_*(m)neq s(m)setminus {0}$, $...

Let $D$ be a weighted oriented graph and $I(D)$ its edge ideal. If contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary sufficient for the equality ordinary symbolic powers ideal having each in some it. characterize naturally unicyclic graphs with unique cycles even their ideals. D^{\prime} obtained from afte...

Let D be a weighted oriented graph, whose underlying graph is G , and let I (D) its edge ideal. If has no 3-, 5-, or 7-cycles, K?nig, we characterize when unmixed. 3- 5-cycles, Cohen–Macaulay. We prove that unmixed if only Cohen–Macaulay girth greater than 7 K?nig 4-cycles.

To an arbitrary directed graph we associate a row-finite directed graph whose Calgebra contains the C-algebra of the original graph as a full corner. This allows us to generalize results for C-algebras of row-finite graphs to C-algebras of arbitrary graphs: the uniqueness theorem, simplicity criteria, descriptions of the ideals and primitive ideal space, and conditions under which a graph algeb...

We introduce a new free, open-source compound graph editing and layout framework named Chisio, based on the Eclipse Graph Editing Framework (GEF) and written in Java. Chisio can be used as a finished graph editor with its easy-to-use graphical interface. The framework has an architecture suitable for easy customization of the tool for end-user’s specific needs as well. Chisio comes with a varie...

Abstract We show that every graded ideal of a Leavitt path algebra is isomorphic to algebra. It known I the graph, as generalised hedgehog which defined based on certain sets vertices uniquely determined by . However, this isomorphism may not be graded. replacing short ‘spines’ graph with possibly fewer, but then necessarily longer spines, we obtain (which call porcupine graph) whose Our proof ...