Dlab-Ringel's standardization method gives a realization of a standardly stratified algebra. In this paper, we construct mixed stratified algebras, which are a generalization of standardly stratified algebras, following Dlab-Ringel's standardization method. Moreover, we study a Ringel duality of mixed stratified algebras from the viewpoint of stratifying systems.

Ringel's right-strongly quasi-hereditary algebras are a distinguished class of quasi-hereditary algebras of Cline-Parshall-Scott. We give characterizations of these algebras in terms of heredity chains and right rejective subcategories. We prove that any artin algebra of global dimension at most two is right-strongly quasi-hereditary. Moreover we show that the Auslander algebra of a representation-finite algebra A is strongly quasi-hereditary if and only if A is a Nakayama algebra.2010 Mathematics Subject Classification. 16G10, 18A40.

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As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalises hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: one is the bijection induced by the HRS-tilt of t-structures on triangulated categories. The other is Asai–Pfeifer’s and Tattar’s bijections for torsion pairs in an abelian category, which is related to $\tau$ -tilting reduction and brick labelling.

As a general framework for the studies of t-structures on triangulated categories and torsion pairs in abelian categories, we introduce the notions of extriangulated categories with negative first extensions and s-torsion pairs. We define a heart of an interval in the poset of s-torsion pairs, which naturally becomes an extriangulated category with a negative first extension. This notion generalizes hearts of t-structures on triangulated categories and hearts of twin torsion pairs in abelian categories. In this paper, we show that an interval in the poset of s-torsion pairs is bijectively associated with s-torsion pairs in the corresponding heart. This bijection unifies two well-known bijections: One is the bijection induced by HRS-tilt of t-structures on triangulated categories. The other is Asai-Pfeifer's and Tattar's bijections for torsion pairs in an abelian category, which is related to τ -tilting reduction and brick labeling.Recently, inspired by τ -tilting theory [AIR], the study of the poset structures of torsion pairs in abelian categories has been developed by various authors [IRTT, BCZ, DIRRT, AP]. Let t 1 := (T 1 , F 1 ) and t 2 := (T 2 , F 2 ) be torsion pairs in an abelian category A. We define t 1 ≤ t 2 if it satisfies T 1 ⊆ T 2 , and in this case, [t 1 , t 2 ] denotes the interval in the poset of torsion pairs in A consisting of t with t 1 ≤ t ≤ t 2 . We call the subcategorywhich tells us "the difference" between t 1 and t 2 . As for this heart, the following isomorphism was established in [AP] (when the heart is a wide subcategory) and [T].Theorem 1.2. [T, Theorem A] Let A be an abelian category and [t 1 , t 2 ] an interval in the poset of torsion pairs in A. Then there exists a poset isomorphism between [t 1 , t 2 ] and the poset of torsion pairs inThis isomorphism originally appeared in the context of τ -tilting reduction in [Ja], and is a useful tool to study various subcategories of module categories, e.g. [DIRRT, AP, ES].The aim of this paper is to show that two poset isomorphisms in Theorem 1.1 and Theorem 1.2 are consequences of a more general poset isomorphism in extriangulated categories. We