We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets, but gives a more finely graded lower bound. These problems are solved for certai...

generalizing the concepts of -fuzzy (left, right, lateral) ideals, -fuzzy quasi-ideals and   -fuzzy bi (generalized bi-) ideals in ternary semigroups, the notions of -fuzzy (left, right, lateral) ideals, -fuzzy quasi-ideals and -fuzzy bi (generalized bi-) in ternary semigroups are introduced and several related properties are investigated. some new results are obtained.

Let be a root system with a subset of positive roots, +. We consider edges of the Hasse diagrams of some posets associated with +. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For +, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple root. We give several descriptions of the se...

In this paper, we provide the necessary and sufficient conditions for edge binomials of tree forming a [Formula: see text]-sequence in terms degree sequence notion graph. We study regularity powers binomial ideals trees generated by binomials.

In this paper, we study depth and Stanley of the edge ideals quotient rings ideals, associated with classes graphs obtained by strong product two graphs. We consider cases when either both are arbitrary paths or one is an path other cycle. give exact formula for values some subclasses. also sharp upper bounds in general cases.

Consider edge colorings of directed graphs where edges of the form v1v2 and v2v3 must have different colors. Here, v1 ≠ v2 , v2 ≠ v3 but v1 = v3 is possible. It is known that this coloring induces a vertex coloring by sets of edge colors, in which edge v1v2 in the graph implies that the set color of v1 contains an element not in the set color of v2; conversely, each such set coloring of vertice...

The monomial initial ideals of a graded polynomial ideal are in bijection with the vertices of a convex polytope known as the state polytope of the ideal. The Grr obner fan of the ideal is the normal fan of its state polytope. In this paper we present a software system called TiGERS (Toric Grr obner bases Enumeration by Reverse Search) for computing the Grr obner fan of a toric ideal by enumera...

let  $r$ be a commutative noetherian ring and $i$ be an ideal of $r$. we say that $i$ satisfies the persistence property if  $mathrm{ass}_r(r/i^k)subseteq mathrm{ass}_r(r/i^{k+1})$ for all positive integers $kgeq 1$, which $mathrm{ass}_r(r/i)$ denotes the set of associated prime ideals of $i$. in this paper, we introduce a class of square-free monomial ideals in the polynomial ring  $r=k[x_1,ld...