Mathematics

Let C be a class of Hausdorff topological semigroups which contains all zero-dimensional Hausdorff topological semigroups. A semigroup X is called C - closed if X is closed in each topological semigroup Y?�C containing X as a discrete subsemigroup; X is projectively C - closed if for each congruence ??on X the quotient semigroup X / ??is C -closed. A semigroup X is called chain - finite if for any infinite set I?�X there are elements x,y?�I such that xy?�{x,y} . We prove that a semigroup X is C -closed if it admits a homomorphism h:X?�E to a chain-finite semilattice E such that for every e?�E the semigroup h ?? (e) is C -closed. Applying this theorem, we prove that a commutative semigroup X is C -closed if and only if X is periodic, chain-finite, all subgroups of X are bounded, and for any infinite set A?�X the product AA is not a singleton. A commutative semigroup X is projectively C -closed if and only if X is chain-finite, all subgroups of X are bounded and the union H(X) of all subgroups in X has finite complement X?�H(X) .

In this paper, we investigate containment statements between symbolic and ordinary powers and bounds on the Waldschmidt constant of defining ideals of points in projective spaces. We establish the stable Harbourne conjecture for the defining ideal of a general set of points. We also prove Chudnovsky's Conjecture and the stable version of the Harbourne--Huneke containment conjectures for a general set of sufficiently many points.

Let ? be a numerical semigroup and I?��?be an ideal of ? . The graph G I (?) assigned to an ideal I of ? is a graph with elements of (??�I ) ??as vertices and any two vertices x,y are adjacent if and only if x+y?�I . In this paper we give a complete characterization (up to isomorphism ) of the graph G I (?) to be planar, where I is an irreducible ideal of ? . This will finally characterize non planar graphs G I (?) corresponding to irreducible ideal I .

Generalizing the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC), a class of constraints involving a pair of operators was considered in \cite{ZGK2}. For a given constraint, the existences of extensions of differential and Rota-Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are shown to be equivalent. In this paper, we give a classification of the constraints satisfying these equivalent conditions.

Nearly complete intersection ideals were introduced by A. Boocher and J. Seiner (2018) and defines a special class of monomial ideals in a polynomial ring. These ideals were used to give a lower bound of the total sum of betti numbers that appear a minimal free resolution of a monomial ideal. In this note we give a graph theoretic classification of nearly complete intersection ideals generated in degree two. In doing so, we define a novel graph operation (the inversion) that is motivated by the definition of this new class of ideals.

An element in a ring R is called clear if it is the sum of unit-regular element and unit. An associative ring is clear if every its element is clear. In this paper we defined clear rings and extended many results to wider class. Finally, we proved that a commutative Bézout domain is an elementary divisor ring if and only if every full matrix order 2 over it is nontrivial clear.

We introduce a family of squarefree monomial ideals associated to finite simple graphs, whose monomial generators correspond to closed neighborhood of vertices of the underlying graph. Any such ideal is called the closed neighborhood ideal of the graph. We study some algebraic invariants of these ideals like Castelnuovo-Mumford regularity and projective dimension and present some combinatorial descriptions for these invariants in terms of graph invariants.

The cut sets of a graph are special sets of vertices whose removal disconnect the graph. They are fundamental in the study of binomial edge ideals, since they encode their minimal primary decomposition. We introduce the class of accessible graphs as the graphs with unmixed binomial edge ideal and whose cut sets form an accessible set system. We prove that the graphs whose binomial edge ideal is Cohen-Macaulay are accessible and we conjecture that the converse holds. We settle the conjecture for large classes of graphs, including chordal and traceable graphs, providing a purely combinatorial description of Cohen-Macaulayness. The key idea in the proof is to show that both properties are equivalent to a further combinatorial condition, which we call strong unmixedness.

We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs which have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by Pitones, Reyes and Toledo.

Given a standard graded polynomial ring over a commutative Noetherian ring A , we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when A is a field of characteristic zero, and follows from a result of Peskine and Szpiro when A is a field of positive characteristic; our result applies, for example, when A is the ring of integers.