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Monoidal logics were introduced as a foundational framework to analyze the proof theory of logical systems. Inspired by Lambek's seminal work in categorical logic, the objective is to define logical systems in order to make explicit their categorical (monoidal) structure. In this setting, logical connectives can be proven to be functors with specific properties. Accordingly, monoidal logics allow a classification of logical systems in function of their categorical structure and the functorial properties of their connectives. As they stand, however, strong parallels can be made between monoidal logics and the broader proof-theoretical framework of display logics. In this paper, we extend the results presented in Peterson ((2016). A comparison between monoidal and substructural logics. Journal of Applied Non-Classical Logics, 26(2), 126–159) and we show that monoidal logics are sound and complete with respect to associative display logics, thus providing a completeness result with regards to the algebraic semantics of display and substructural logics. In addition, we discuss the notions of classical and intuitionistic systems. Starting from Lambek's and Grishin's analyses, we explore the role played by partial De Morgan dualities and discuss the necessary and sufficient conditions required for the definitions of classical and intuitionistic deductive systems. [ABSTRACT FROM AUTHOR]

In this paper, we firstly give some basic properties on quasi-cotilting modules. With the help of these properties, we obtain a Quasi-cotilting Theorem (see Theorem 2.8). In particular, the Cotilting Theorem in [5] is a corollary of our result. Finally, we introduce a new notion, hereditary quasi-cotilting modules, and mainly prove that the class consisting of all Δ-reflexive modules for a hereditary quasi-cotilting module is closed under submodules. [ABSTRACT FROM AUTHOR]

In this paper, we study lattices of preradicals which are not small classes (in which case we say that the corresponding rings arep-large), and specially we consider some infinite representation type algebras. We construct an injective assignment between lattices of preradicals, using a full functor between the corresponding categories of modules, that satisfies certain conditions. We show that the polynomial ring over any field isp-large, and we use this fact to provide examples and some classes of algebras (both tame and wild) which arep-large. [ABSTRACT FROM PUBLISHER]

Bulman-Fleming et al. [1] introduced the properties flatness, (principal) weak flatness and torsion freeness ofS-posets over pomonoidsS, and presented the implication: “principal weak flatness ⇒ torsion freeness”. Golchin et al. [3] described flatness, weak flatness and principal weak flatness of rightS-posetsASover a pomonoidSin terms of pullback preservation by the tensor multiplication functorAS⊗−. In this note we first show, by giving a counterexample, that this implication of Bulman-Fleming et al. is incorrect. Afterwards we provide examples that deny these descriptions of Golchin et al. In view of the above, we finally study another property ofS-posets (we call it weak torsion freeness), which is weaker than principal weak flatness, and characterize pomonoids by this property. [ABSTRACT FROM AUTHOR]

Recently, the notion of Gorenstein AC-projective (resp., Gorenstein AC-injective) modules was introduced in [3] by which the so-called “Gorenstein AC-homological algebra” was established. Here, we define and study a notion of Gorenstein AC-projective dimension for complexes (not necessarily bounded) over associative rings, which is inspired by Veliche’s construction of defining Gorenstein projective dimension. In particular, we show that such a dimension can be closely related to the “proper” Gorenstein AC-projective resolutions of complexes induced by a complete and hereditary cotorsion pair in the category of complexes of modules. This enables us to interpret this dimension of a complex in terms of vanishing of the derived functorRHomR(−,−). As applications, some characterizations of the Gorenstein AC-projective dimension of a module are also obtained. [ABSTRACT FROM PUBLISHER]

We study the functorial properties of Bridgeland’s Hall algebras. Specifically, let𝒜andℬbe two categories satisfying certain conditions for the definitions of Bridgeland’s Hall algebras, and letF:𝒜→ℬbe a fully faithful exact functor, which preserves projectives, thenFinduces an embedding of algebras from the Bridgeland’s Hall algebra of𝒜to the one ofℬ. In addition, letAbe a finite-dimensional algebra over a finite field andBsome special quotient algebra ofA, then the Bridgeland’s Hall algebra ofBis the quotient algebra of the one ofA. Moreover, we consider the BGP-reflection functors on the category of 2-cyclic complexes and obtain some homomorphisms of algebras among the subalgebras of Bridgeland’s Hall algebras. [ABSTRACT FROM PUBLISHER]

In this article, we study some algebraic and combinatorial behaviors of expansion functor. We show that on monomial ideals some properties like polymatroidalness, weakly polymatroidalness, and having linear quotients are preserved under taking the expansion functor. The main part of the article is devoted to study of toric ideals associated to the expansion of subsets of monomials which are minimal with respect to divisibility. It is shown that, for a given discrete polymatroidP, if toric ideal ofPis generated by double swaps, then toric ideal of any expansion ofPhas such a property. This result, in a special case, says that White's conjecture is preserved under taking the expansion functor. Finally, the construction of Gröbner bases and some homological properties of toric ideals associated to expansions of subsets of monomials is investigated. [ABSTRACT FROM PUBLISHER]

Using Bak's localization-completion method and a version of Stein's relativization, we obtain a descendingS0U2n(R, 𝔏)-central seriesS0U2n(R, 𝔏, ℑ) ≥ S1U2n(R, 𝔏, ℑ) ≥ S2U2n(R, 𝔏, ℑ) ≥ …, such thatSdU2n(R, 𝔏, ℑ) coincides withEU2n(R, 𝔏, ℑ) wheneverd ≥ Bass–Serre dimension ofR. Moreover, eachSdU2n(R, 𝔏, ℑ) is normal infor anyd ≥ 0. [ABSTRACT FROM PUBLISHER]

A notion of mutation pairs of subcategories in an abelian category is defined in this article. For an extension closed subcategory 𝒵 and a rigid subcategory 𝒟 ⊂ 𝒵, the subfactor category 𝒵/[𝒟] is also a triangulated category whenever (𝒵, 𝒵) forms a 𝒟-mutation pair. Moreover, if 𝒟 and 𝒵 satisfy certain conditions in modΛ, the category of finitely generated Λ-modules over an artin algebra Λ, the triangulated category 𝒵/[𝒟] has a Serre functor. [ABSTRACT FROM PUBLISHER]

This paper is an attempt to solve the following problem: given a logic, how to turn it into a paraconsistent one? In other words, given a logic in whichex falso quodlibetholds, how to convert it into a logic not satisfying this principle? We use a framework provided by category theory in order to define a category of consequence structures. Then, we propose a functor to transform a logic not able to deal with contradictions into a paraconsistent one. Moreover, we study the case of paraconsistentization of propositional classical logic. [ABSTRACT FROM AUTHOR]

We provide a counterexample to a theorem of Ford, namely a pushout square of categories with all involved functors injective, such that there is no associated exact “Mayer–Vietoris” sequence of derived limits. Further, we construct a Mayer–Vietoris sequence for derived (co)limits under some additional hypotheses, extending the well-known case of a pushout square of group monomorphisms. [ABSTRACT FROM PUBLISHER]

We prove that, ifF,G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts andUis a generator of 𝒞, then there is a bijection between Nat(F,G) and the centralizer of Hom𝒟(F(U),G(U)) considered as an Hom𝒞(U,U)-Hom𝒞(U,U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories. [ABSTRACT FROM PUBLISHER]

It is well known that the abelianization of a groupGcan be computed as the cokernel of the diagonal morphism (1G, 1G):G → G × Gin the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism. [ABSTRACT FROM PUBLISHER]

In this article, we study the structure and representahility of the automorphism g¡roup functor of the V = 4 Lie conforma! superalgebra over an algebraically closed field k of characteristic zero. [ABSTRACT FROM AUTHOR]

We find a new representation of the simple Lie algebra of typeE7on the polynomial algebra in 27 variables. Using this representation and Shen's idea of mixed product, we construct a new functor fromE6-ModtoE7-Mod. A condition for the functor to map a finite-dimensional irreducibleE6-module to an infinite-dimensional irreducibleE7-module is obtained. Our general framework also gives a direct polynomial extension from irreducibleE6-modules to irreducibleE7-modules, which can be used to derive Gel'fand–Zetlin bases forE7from those forE6that can be obtained from those forD5according to our earlier work. [ABSTRACT FROM AUTHOR]

LetAbe a differential graded (DG) algebra with a trivial differential over a commutative unital ring. This paper investigates the image of the totaling functor, defined from the category of complexes of gradedA-modules to the category of DGA-modules. Specifically, we exhibit a special class of semifree DGA-modules which can always be expressed as the totaling of some complex of graded freeA-modules. As a corollary, we also provide results concerning the image of the totaling functor whenAis a polynomial ring over a field. [ABSTRACT FROM AUTHOR]

The article focuses on continuity for functions between topological spaces as an adjointness condition between naturally arising functors. Topics discussed includes category theory, introduced by S. Eilenberg and S. Mac Lane, impact of the category theory on algebra, geometry and mathematical physics and relationships among different categories established via functors between them.

The manuscript makes a category-theoretic comparison between the concepts of topological system of S. Vickers and attachment relation of C. Guido. With the help of the recent definition of the notions in the framework of an arbitrary variety of algebras, we consider functorial relationships between the categories of such generalized structures, and the link of the obtained functors to both spatialization and localification procedure introduced by S. Vickers for topological systems. [ABSTRACT FROM AUTHOR]

Let H be a finite-dimensional Hopf algebra over a field k and A/B be a right H-Galois extension. If the functors A ⊗B − andB(−) are both separable, then the finitistic dimension of A is equal to that of B. [ABSTRACT FROM AUTHOR]

We study Gorenstein right derived functors of − ⊗ −with respect to semidualizing modules. As applications, some new criteria for a semidualizing module to be dualizing are given provided thatRis a ring with a dualizing complex. [ABSTRACT FROM AUTHOR]

We study Hom-type analogs of Rota–Baxter and dendriform algebras, called Rota–BaxterG-Hom–associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota–Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed. [ABSTRACT FROM PUBLISHER]

Some criteria for graded local cohomology to commute with coarsening functors are proven, and an example is given where graded local cohomology does not commute with coarsening. [ABSTRACT FROM AUTHOR]

For a Hopf algebraHover a commutative ringkand a leftH-moduleV, the tensor functors − ⊗kVandV ⊗k − are known to be left adjoint to some kind of Hom-functors as endofunctors ofH𝕄. The units and counits of adjunctions, in this case, are formally trivial as in the classical case. In this paper, we generalize this Hom-tensor adjunction for (bi-)module categories over a quasi-Hopf algebraHand show that these (bi-)module categories are biclosed monoidal. However, the units and counits of adjunctions in these generalized cases are not as trivial as in the Hopf algebra case, and they should be modified in terms of the reassociator and the quasi-antipode. Also, if theH-moduleVis finitely generated and projective as ak-module, we will obtain a generalized form of adjunction between the tensor functors − ⊗Vand − ⊗V* depending on the reassociator and quasi-antipode ofHand describe a natural isomorphism between functors − ⊗V* and Homk(V, −) explicitly. Furthermore, we consider the special caseV = Abeing anH-module algebra. In this case, each tensor functor will be a monad and its corresponding right adjoint is a comonad. We describe isomorphisms between the (Eilenberg–Moore) module categories over these monads and the (Eilenberg–Moore) comodule categories over their corresponding comonads explicitly. [ABSTRACT FROM AUTHOR]

In the paper we describe the subcategory of the category of ℤ-graded Lie algebras which is equivalent to the category of Jordan pairs via a functorial modification of the Tits, Kantor, and Koecher TKK construction. For instance, we prove thatL = L−1⊕L0⊕L1can be constructed from a Jordan pair if and only ifL0 = [L−1,L1] and the second graded homology groupis trivial. Similar descriptions are obtained for Jordan triple systems and Jordan algebras. New functorial versions of the TKK construction are given for pairs and algebras. [ABSTRACT FROM AUTHOR]

The results of [7] and [2] gave a recursive construction for all quasi-hereditary and standardly stratified algebras starting with local algebras and suitable bimodules. Using the notion of stratifying pairs of subcategories, introduced in [3], we generalize these earlier results to construct recursively all CPS-stratified algebras. [ABSTRACT FROM AUTHOR]

We study the existence of covers and envelopes by some special functors on the category of finitely presented modules. As an application, we characterize some important rings using these functors. We also investigate homological properties of some functors on the stable module category. The relationship between phantoms and Ext-phantoms is obtained. It is shown that every leftR-moduleMhas an Ext-phantom preenvelopef:M → Nwith coker(f) pure-projective. Finally, we prove that, as a torsionfree class of (mod-R, Ab), (mod-R, Ab) is generated by theFP-injective objects. [ABSTRACT FROM AUTHOR]

Let A be a finite-dimensional Gorenstein algebra of finite CM-type. We show that the Gorenstein derived category of A has Auslander–Reiten triangles. In this case, we otain a Serre functor in . We also establish a relationship between the Nakayama functors and the Gorenstein derived equivalences. [ABSTRACT FROM PUBLISHER]

We study the structure of infinitely generated small modules over abelian regular rings, i.e., modules over which the covariant functor Hom commutes with direct sums. It is shown that every infinitely generated small module has either an infinitely generated factor which is at most 22ω -generated or a countably generated essential submodule. As a consequence, we prove a module-theoretic criterion of steadiness for abelian regular rings. [ABSTRACT FROM PUBLISHER]

We address four problems regarding representable functors and give answers to them for functors connecting the category of comodules over a coring to the category of modules over a ring. A functor's property of being Frobenius is restated as a particular case of its representability by imposing the predefinition of the object of representability. Let R, S be two rings, C an R-coring and the category of left C-comodules. The category of all representable functors is shown to be equivalent to the opposite of the category . For U an (S, R)-bimodule we give necessary and sufficient conditions for the induction functor to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings. [ABSTRACT FROM PUBLISHER]

We show that the Hilbert functor of points on an arbitrary separated algebraic stack is an algebraic space. We also show the algebraicity of the Hilbert stack of points on an algebraic stack and the algebraicity of the Weil restriction of an algebraic stack along a finite flat morphism. For the latter two results, no separation assumptions are necessary. [ABSTRACT FROM AUTHOR]

Given a complete, cocomplete category C, we investigate the problem of describing those small categories I such that the diagonal functor Δ: C → Functors(I, C) is a Frobenius functor. This condition can be rephrased by saying that the limits and the colimits of functors I → C are naturally isomorphic. We find necessary conditions on I for a certain class of categories C, and, as an application, we give both necessary and sufficient conditions in the two special cases C =Set or RM, the category of left modules over a ring R. [ABSTRACT FROM AUTHOR]

We prove that both the embedding of the category of Hopf algebras into that of bialgebras and the forgetful functor from the category of Hopf algebras to the category of algebras have right adjoints; in other words, every bialgebra has a Hopf coreflection, and on every algebra there exists a cofree Hopf algebra. In this way, we give an affirmative answer to a forty-years old problem posed by Sweedler. On the route, the coequalizers and the coproducts in the category of Hopf algebras are explicitly described. [ABSTRACT FROM AUTHOR]

The paper provides another approach to the notion of variable-basis topological system generalizing the fixed-basis concept of S. Vickers, considers functorial relationships between the categories of modified variable-basis topological systems and variable-basis fuzzy topological spaces in the sense of S.E. Rodabaugh and shows that the procedure of localification is possible in the new setting. [ABSTRACT FROM AUTHOR]

We prove that the functor of tensor multiplication by a right S-poset (S is a pomonoid) preserves all small weighted limits if and only if this S-poset is cyclic and projective. We also show that this functor preserves all finite pie-limits if and only if the S-poset is a filtered colimit of S-posets isomorphic to SS. [ABSTRACT FROM AUTHOR]

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein-Gelfand-Gelfand correspondence. [ABSTRACT FROM AUTHOR]

We investigate functors between abelian categories having a left adjoint and a right adjoint that are similar (these functors are called quasi-Frobenius functors). We introduce the notion of a quasi-Frobenius bimodule and give a characterization of these bimodules in terms of quasi-Frobenius functors. Some applications to corings and graded rings are presented. In particular, the concept of quasi-Frobenius homomorphism of corings is introduced. Finally, a version of the endomorphism ring Theorem for quasi-Frobenius extensions in terms of corings is obtained. [ABSTRACT FROM AUTHOR]

Let R be a ring with 1 and M a right R-module. In this article, we will see that the functor F = M ⊗R - gives rise to a complete hereditary cotorsion pair where the left class consists of the F-acyclic objects. This cotorsion pair induces a Quillen model structure on Ch(R) which recovers the derived functors [image omitted]. An F-acyclic resolution is as good as a cofibrant replacement in this model structure. So in short, we formalize the fact that [image omitted] can be computed using F-acyclic resolutions. [ABSTRACT FROM AUTHOR]

We introduce the notion of “bi-monoid” in general monoidal category, generalizing by this the notion of “bialgebra”. In the case of bimodules over a noncommutative algebra, we obtain compatibility conditions between rings and corings whenever both structures admit the same underlying bimodule. Several examples are expounded in this case. We also show that there is a class of right modules over a bi-monoid which is a monoidal category and the forgetful functor to the ground category is a strict monoidal functor. [ABSTRACT FROM AUTHOR]

In the early 1970s, Campbell published three papers on majorization-subordination results for locally univalent functions. In particular, he showed that if F is linearly invariant of order α and if f is subordinate to F on {z ≔ |z| < 1}, then f ' is majorized by F ' on {z ≔ |z| < m(α)} where [image omitted], provided α ≥ 1.65. He conjectured, in fact, that this result also held for 1.65 > α ≥ 1. We review Campbell's proof and why the restriction α ≥ 1.65 arose in the proof. We then affirmatively verify Campbell's conjecture in Theorem 1. [ABSTRACT FROM AUTHOR]

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These categories naturally fit into a framework of relative derived categories, and once we prove that there are decent resolutions of complexes, we are able to prove many familiar results in homological algebra. [ABSTRACT FROM AUTHOR]

In this note, we introduce and study the notion of sheaf on a noncommutative topology and construct an associated sheafification functor in this noncommutative context. It appears that abstract localization in the category of presheaves on a partially ordered set provides an elegant way to define this sheafification functor, generalizing similar constructions in the commutative case. [ABSTRACT FROM AUTHOR]

A method is developed in which an analytical solution is obtained for certain classes of second-order differential equations with variable coefficients. By the use of transformations and by repeated iterated integration, a desired solution is obtained. This alternative method represents a different way to acquire a solution from classic power series techniques and other approaches. It is, at times, more involved than traditional methods. [ABSTRACT FROM AUTHOR]

We investigate adjoint and Frobenius pairs between categories of comodules over rather general corings. We particularize to the case of the adjoint pair of functors associated to a morphism of corings over different base rings, which leads to a reasonable notion of Frobenius coring extension. When applied to corings stemming from entwining structures, we obtain new results in this setting and in graded ring theory. [ABSTRACT FROM AUTHOR]

In this article we prove that in prime characteristic there is a functor - p-Leib from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover, we define the notion of restricted universal enveloping diassociative algebra Udp(&gfr;) of a restricted Leibniz algebra &gfr; and we show that Udp is left adjoint to the functor - p-Leib. We also construct the restricted enveloping algebra, which classifies the restricted Leibniz modules. In the last section we put a restricted pre-Lie structure on the tensor product of a Leibniz algebra by a Zinbiel algebra. [ABSTRACT FROM AUTHOR]

Using the Stroh formalism combined with the analytical continuation principle of Muskhelishvili, the Green's functions for a line heat source in a piezoelectric solid with a parabolic boundary are obtained in closed form. The obtained Green's functions not only satisfy all the given boundary conditions, but also ensure the displacement and electric potential to be single-valued. As special cases, the solutions for a piezoelectric half-plane are also presented, and they are shown to be consistent with previous works. [ABSTRACT FROM AUTHOR]

In this article we further study the full subcategories of the category of finitely generated modules over an Artin algebra introduced in Platzeck and Pratti (2000), consisting of the modules having an add M resolution of length i , which remains exact under the functor Hom A ( M , -). In particular, we characterize tilting modules in terms of these categories and determine when the transpose of a tilting module is a tilting module. [ABSTRACT FROM AUTHOR]

The class of mixed groups &Gscr; was introduced by Glaz and Wickless. This article investigates subclasses &Cscr; of &Gscr; such that the functor Hom( A , -)/ t Hom( A , -) between &Cscr; and the category of right E ( A )/ tE ( A )-modules is full. [ABSTRACT FROM AUTHOR]

Let S be a partially ordered monoid, or briefly, pomonoid. A right S-poset (often denoted AS) is a poset A together with a right S-action (a, s) as that is monotone in both arguments and that satisfies the conditions a(st) = (as)t and a1 = 1 for all a ∈ A, s, t ∈ S. Left S-posets SB are defined analogously, and the left or right S-posets form categories, S-POS and POS-S, whose morphisms are the monotone maps that preserve the S-action. In these categories, as in the category POS of posets, the monomorphisms and epimorphisms are the injective and surjective morphisms, respectively, but the embeddings and quotient maps have stronger properties; in particular, an embedding is a monomorphism that is also an order embedding. A tensor product AS ⊗ SB exists (a poset) that has the customary universal property with respect to balanced, bimonotone maps from A × B into posets. Various flatness properties of AS can be defined in terms of the functor AS ⊗− from S-POS into POS. More specifically, an S-poset AS is called flat if the induced morphism AS ⊗ SB → AS ⊗ SC is injective whenever SB → SC is an embedding in S-POS this means that, for all SB and all a, a' ∈ A and b, b' ∈ B, if a ⊗ b = a' ⊗ b' in A ⊗ B, then the same equality holds in A ⊗ (Sb ∪ Sb'). AS is called (principally) weakly flat if the induced morphism above is injective for all embeddings of (principal) left ideals into SS. Similarly, AS is called po-flat if the functor AS ⊗− preserves embeddings: for this definition, replace = by ≤ in the description above (see Shi, 2005). Weak and principally weak versions of po-flatness are defined in an obvious way. In the present article, we first consider flatness properties for one-element and Rees factor S-posets. We present examples that distinguish between various types of flatness and the corresponding, generally stronger, notions of po-flatness. We then initiate a study of absolute flatness for pomonoids: a monoid (respective pomonoid) S is called right absolutely flat if all right S-acts (respective S-posets) are flat. The findings for absolute flatness of pomonoids are markedly different from the corresponding unordered results. [ABSTRACT FROM AUTHOR]

An effective method to compute a presentation of Tor k ( M , N ) for modules on a not necessarily commutative algebra is proposed. [ABSTRACT FROM AUTHOR]

It is known from Auslander et al. (1979) and Ringel (1990) that the Bernstein--Gelfand--Ponomarev reflection functors are special cases of tilting functors, and these reflection functors induce isomorphisms between certain subalgebras of Hall algebras. In this paper we generalize this result by proving that the tilting functors, in general, also induce isomorphisms between certain subalgebras of Hail algebras. [ABSTRACT FROM AUTHOR]