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Let $T$ be a right exact functor from an abelian category $mathscr{B}$ into another abelian category $mathscr{A}$. Then there exists a functor ${bf p}$ from the product category $mathscr{A}timesmathscr{B}$ to the comma category $(Tdownarrowmathscr{A})$. In this paper, we study the property of the extension closure of some classes of objects in $(Tdownarrowmathscr{A})$, the exactness of the functor ${bf p}$ and the detail description of orthogonal classes of a given class ${bf p}(mathcal{X},mathcal{Y})$ in $(Tdownarrowmathscr{A})$. Moreover, we characterize when special precovering classes in abelian categories $mathscr{A}$ and $mathscr{B}$ can induce special precovering classes in $(Tdownarrowmathscr{A})$. As an application, we prove that under suitable cases, the class of Gorenstein projective left $Lambda$-modules over a triangular matrix ring $Lambda=left(begin{smallmatrix}R & M O & S end{smallmatrix} right)$ is special precovering if and only if both the classes of Gorenstein projective left $R$-modules and left $S$-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
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Let $t$ be a positive real number. A graph is called $t$-tough, if the removal of any cutset $S$ leaves at most $|S|/t$ components. The toughness of a graph is the largest $t$ for which the graph is $t$-tough. A graph is minimally $t$-tough, if the toughness of the graph is $t$ and the deletion of any edge from the graph decreases the toughness. In this paper we investigate the minimum degree and the recognizability of minimally $t$-tough graphs in the class of chordal graphs, split graphs, claw-free graphs and $2K_2$-free graphs.
Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is $Theta_2^{text{P}}$-complete. They studied the common graph parameters $alpha$ (independence number), $beta$ (vertex cover number), $omega$ (clique number), and $chi$ (chromatic number) for certain variants of the stability problem. We follow their approach and provide a large number of polynomial-time algorithms solving these problems for special graph classes, namely for graphs without edges, complete graphs, paths, trees, forests, bipartite graphs, and co-graphs.
In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
Comtrans algebras, arising in web geometry, have two trilinear operations, commutator and translator. We determine a Grobner basis for the comtrans operad, and state a conjecture on its dimension formula. We study multilinear polynomial identities for the special commutator $[x,y,z] = xyz-yxz$ and special translator $langle x, y, z rangle = xyz-yzx$ in associative triple systems. In degree 3, the defining identities for comtrans algebras generate all identities. In degree 5, we simplify known identities for each operation and determine new identities relating the operations. In degree 7, we use representation theory of the symmetric group to show that each operation satisfies identities which do not follow from those of lower degree but there are no new identities relating the operations. We use noncommutative Grobner bases to construct the universal associative envelope for the special comtrans algebra of $2 times 2$ matrices.
Let $mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $operatorname{Hom}$ and $operatorname{Ext}$ spaces. It is proved that the bounded derived category $mathcal{D}^b(mathcal{H})$ has a silting object iff $mathcal{H}$ has a tilting object iff $mathcal{D}^b(mathcal{H})$ has a simple-minded collection with acyclic $operatorname{Ext}$-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $mathcal{D}^b(mathcal{H})$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $mathcal{R}$ of $mathcal{D}^b(mathcal{H})$ can be completed into a simple-minded collection iff the $operatorname{Ext}$-quiver of $mathcal{R}$ is acyclic.
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