Relative tilting pairs

In this paper, we introduce and study (weak) pure-injective Gorenstein projective modules. Let [Formula: see text] be an Artin algebra. We prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules coincides with the intersection of the category of pure-injective left [Formula: see text]-modules and that of Gorenstein projective left [Formula: see text]-modules. Then, we get an equivalent characterization of virtually Gorenstein algebras (being CM-finite). Furthermore, we prove that the category of weak pure-injective Gorenstein projective left [Formula: see text]-modules is enveloping in the category of left [Formula: see text]-modules; and if [Formula: see text] is virtually Gorenstein, then it is precovering in the category of pure-injective left [Formula: see text]-modules.

Let be an Artin algebra. Under certain Auslander‐type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander‐type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if is left quasi‐Auslander, then is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if satisfies the Auslander condition, then is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander–Reiten's conjecture, which states that is Gorenstein if satisfies the Auslander condition.

Let R and S be rings, and let RωS be a semidualizing bimodule. We prove that there exists a Morita equivalence between the class of ∞-ω-cotorsion-free modules and a subclass of the class of ω-adstatic modules. Also, we establish the relation between the relative homological dimensions of a module M and the corresponding standard homological dimensions of Hom(ω,M). By investigating the properties of the Bass injective dimension of modules (resp., complexes), we get some equivalent characterizations of semitilting modules (resp., Gorenstein Artin algebras). Finally, we obtain a dual version of the Auslander–Bridger approximation theorem. As a consequence, we get some equivalent characterizations of Auslander n-Gorenstein Artin algebras.

We introduce an operator on classes of regular languages, the star-free closure. Our motivation is to generalize standard results of automata theory within a unified framework. Given an arbitrary input class \(\mathscr{C}\) , the star-free closure operator outputs the least class closed under Boolean operations and language concatenation, and containing all languages of \(\mathscr{C}\) as well as all finite languages. We establish several equivalent characterizations of star-free closure: in terms of regular expressions, first-order logic, pure future and future-past temporal logic, and recognition by finite monoids. A key ingredient is that star-free closure coincides with another closure operator, defined in terms of regular operations where Kleene stars are allowed in restricted contexts. A consequence of this first result is that we can decide membership of a regular language in the star-free closure of a class whose separation problem is decidable. Moreover, we prove that separation itself is decidable for the star-free closure of any finite class, and of any class of group languages having itself decidable separation (plus mild additional properties). We actually show decidability of a stronger property, called covering.

A unified framework of fuzzy implications and coimplications

A ring extension is a ring homomorphism preserving identities. In this paper, we give the definitions of relative injective envelopes and relative projective covers of modules on ring extensions, and study their basic properties. In particular, we give their equivalent characterizations in terms of relative essential monomorphisms and relative superfluous epimorphisms, and prove that relative injective envelopes and relative projective covers on ring extensions are unique up to isomorphism whenever they exist. Moreover, for an extension of Artin algebras, we show that every finitely generated module has both a relative injective envelope and a relative projective cover. In addition, we compare relative injective envelopes and relative projective covers on two ring extensions linked by surjective homomorphisms of rings respectively.

For a left and right Noetherian ring R , we give some equivalent characterizations for _RR satisfying the Auslander condition in terms of the flat (resp. injective) dimensions of the terms in a minimal injective coresolution (resp. flat resolution) of left R -modules. Furthermore, we prove that for an artin algebra R satisfying the Auslander condition, R is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander–Gorenstein rings and Auslander-regular rings.

We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander–Solberg relative tilting modules on Artin algebras to infinitely generated tilting modules on arbitrary rings. Furthermore, we see that it presents a tool for developing new tilting theories in categories that can be embedded nicely in an abelian category. In particular, we will show how the tilting theory in exact categories built this way, coincides with tilting objects in extriangulated categories introduced recently. We will review Bazzoni’s tilting characterization, the relative homological dimensions on the induced tilting classes and parametrise certain cotorsion-like pairs by using n-X-tilting classes. As an application, we show how to construct relative tilting classes and cotorsion pairs in Rep(Q,C) (the category of representations of a quiver Q in an abelian category C) from tilting classes in C, where Q is finite-cone-shape.

Logic programs with abstract constraint atoms provide a unifying framework for studying logic programs with various kinds of constraints. Establishing strong equivalence between logic programs is a key property for program maintenance and optimization, and for guaranteeing the same behavior for a revised original program in any context. In this paper, we study strong equivalence of logic programs with abstract constraint atoms. We first give a general characterization of strong equivalence based on a new definition of program reduct for logic programs with abstract constraints. Then we consider a particular kind of program revision—constraint replacements addressing the question: under what conditions can a constraint in a program be replaced by other constraints, so that the resulting program is strongly equivalent to the original one.KeywordsLogic ProgramLogic ProgrammingPropositional AtomAdmissible SolutionNormal ProgramThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nested logic programs and epistemic logic programs are two important extensions of answer set programming. However, the relationship between these two formalisms is rarely explored. In this paper we first introduce the epistemic HT-logic, and then propose a more general extension of logic programs called nested epistemic logic programs. The semantics of this extension – named equilibrium views – is defined on the basis of the epistemic HT-logic. We prove that equilibrium view semantics extends both the answer sets of nested logic programs and the world views of epistemic logic programs. Therefore, our work establishes a unifying framework for both nested logic programs and epistemic logic programs. Furthermore, we also provide a characterization of the strong equivalence of two nested epistemic logic programs.

Tachikawa’s second conjecture predicts that a finitely generated, self-orthogonal module over a finite-dimensional self-injective algebra is projective. This conjecture is an important part of the Nakayama conjecture. Our principal motivation of this work is a systematic understanding of finitely generated, self-orthogonal generators over a self-injective Artin algebra from the view point of stable module categories. Consequently, we give equivalent characterizations of Tachikawa’s second conjecture in terms of M M -Gorenstein categories, and establish a recollement of the M M -relative stable categories for a self-orthogonal generator M M . Further, we show that the Nakayama conjecture holds true for Gorenstein-Morita algebras.

A unified framework of 0-overlap functions and 1-grouping functions