We show that the mesh mutations are the minimal relations among the gg‐vectors with
respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result …

The aim of this paper is to develop a framework for localization theory of triangulated
categories\(\mathcal {C}\), that is, from a given extension-closed subcategory\(\mathcal {N}\) …

Extriangulated categories were introduced by Nakaoka and Palu, which unify exact
categories and extension-closed subcategories of triangulated categories, and recently …

In this paper, we study the relative tilting theory in extriangulated categories. We introduce
the notion of relative tilting classes in an extriangulated category and then give some …

This if the final paper in the seriesContinuous Quivers of Type $ A $. In this part, we
generalize existing geometric models of type $ A $ cluster structures to the new $\mathbf E …

We consider the notion of pre-extriangulated categories. We give some characterizations of
almost split $\mathfrak s $-triangles in a pre-extriangulated category and provide necessary …

Let be an extriangulated category with enough projectives P \mathcalP and enough
injectives I \mathcalI, and let be a contravariantly finite rigid subcategory of which contains P …

This if the final paper in the series Continuous Quivers of Type A. In this part, we generalize
existing geometric models of type A cluster structures for the new E-clusters introduced in …

In this article, we study the Gorenstein property of abelian quotient categories induced by
fully rigid subcategories on an exact category B. We also study when d-cluster tilting …

Let B be an extriangulated category with enough projectives and enough injectives. Let C be
a fully rigid subcategory of B which admits a twin cotorsion pair ((C, K),(K, D)). The quotient …