On the complexity of equational decision problems for finite height complemented and orthocomplemented modular lattices

We study the computational complexity of the satisfiability problem and the complement of the equivalence problem for complemented (orthocomplemented) modular lattices L and classes thereof. Concerning a simple L of finite height, $$\mathcal {NP}$$ NP -hardness is shown for both problems. Moreover, both problems are shown to be polynomial-time equivalent to the same feasibility problem over the division ring D whenever L is the subspace lattice of a D-vector space of finite dimension at least 3. Considering the class of all finite dimensional Hilbert spaces, the equivalence problem for the class of subspace ortholattices is shown to be polynomial-time equivalent to that for the class of endomorphism $$*$$ ∗ -rings with pseudo-inversion; moreover, we derive completeness for the complement of the Boolean part of the nondeterministic Blum-Shub-Smale model of real computation without constants. This result extends to the additive category of finite dimensional Hilbert spaces, enriched by adjunction and pseudo-inversion.

Additive properties for the pseudo core inverse of morphisms in an additive category

In this paper, given a morhism [Formula: see text] with its pseudo core inverse [Formula: see text] and a morphism [Formula: see text] such that [Formula: see text] is invertible, a necessary and sufficient condition and two sufficient conditions are presented under which the additive property, namely [Formula: see text] holds. Several interesting results about additive properties of core inverses of bounded linear operators presented in Huang et al. are generalized to the case of pseudo core inverse of morphism. Also, many results regarding additive properties of core-EP inverses of complex matrices studied by Ma and Stanimirović are extended to the cases of morphism.

ARABIC DISCOURSE MARKERS IN AL-JAZEERA.NET NEWS ARTICLE

This study investigates discourse markers (DMs) development and diversity in accordance with the Modern Standard Arabic (MSA) approach employed in online news articles. 867 football news articles were downloaded from Al-Jazeera.net to analyse the use of discourse markers (DMs) based on previous studies along with Modern Standard Arabic grammar books as well as the theories of Fraser (2005) and Al-Khawaldeh (2014). Both data coding and analysis were completed manually. The results showed that in linking textual fragments, discourse markers under the Additive category are the most frequently used within all the articles. In fact, the study also found that DM wa was most frequently used in articles compared to other discourse markers. For each category, the researcher discovered a set of new discourse markers that were not considered in previous works. The manual coding process has also led to the discovery of a new category – Affirmative, with a comprehensive list of discourse markers. It is hoped that this study can function as an important reference for researchers who are interested to delve deeper into the area.

A Constructive Approach to Freyd Categories

We discuss Peter Freyd’s universal way of equipping an additive category $$\mathbf {P}$$ P with cokernels from a constructive point of view. The so-called Freyd category $$\mathcal {A}(\mathbf {P})$$ A ( P ) is abelian if and only if $$\mathbf {P}$$ P has weak kernels. Moreover, $$\mathcal {A}(\mathbf {P})$$ A ( P ) has decidable equality for morphisms if and only if we have an algorithm for solving linear systems $$X \cdot \alpha = \beta $$ X · α = β for morphisms $$\alpha $$ α and $$\beta $$ β in $$\mathbf {P}$$ P . We give an example of an additive category with weak kernels and decidable equality for morphisms in which the question whether such a linear system admits a solution is computationally undecidable. Furthermore, we discuss an additional computational structure for $$\mathbf {P}$$ P that helps solving linear systems in $$\mathbf {P}$$ P and even in the iterated Freyd category construction $$\mathcal {A}( \mathcal {A}(\mathbf {P})^{\mathrm {op}} )$$ A ( A ( P ) op ) , which can be identified with the category of finitely presented covariant functors on $$\mathcal {A}(\mathbf {P})$$ A ( P ) . The upshot of this paper is a constructive approach to finitely presented functors that subsumes and enhances the standard approach to finitely presented modules in computer algebra.

Closing the category of finitely presented functors under images made constructive

For an additive category P we provide an explicit construction of a category Q(P) whose objects can be thought of as formally representing im(γ)im(ρ)∩im(γ) for given morphisms γ:A→B and ρ:C→B in P, even though P does not need to admit quotients or images. We show how it is possible to calculate effectively within Q(P), provided that a basic problem related to syzygies can be handled algorithmically. We prove an equivalence of Q(P) with the smallest subcategory of the category of contravariant functors from P to the category of abelian groups Ab which contains all finitely presented functors and is closed under the operation of taking images. Moreover, we characterize the abelian case: Q(P) is abelian if and only if it is equivalent to fp(Pop,Ab), the category of all finitely presented functors, which in turn, by a theorem of Freyd, is abelian if and only if P has weak kernels.The category Q(P) is a categorical abstraction of the data structure for finitely presented R-modules employed by the computer algebra system Macaulay2, where R is a ring. By our generalization to arbitrary additive categories, we show how this data structure can also be used for modeling finitely presented graded modules, finitely presented functors, and some not necessarily finitely presented modules over a non-coherent ring.

On the Category of Weakly U-Complexes

Motivated by a study of Davvaz and Shabbani which introduced the concept of U-complexes and proposed a generalization on some results in homological algebra, we study thecategory of U-complexes and the homotopy category of U-complexes. In [8] we said that the category of U-complexes is an abelian category. Here, we show that the object that we claimed to be the kernel of a morphism of U-omplexes does not satisfy the universal property of the kernel, hence wecan not conclude that the category of U-complexes is an abelian category. The homotopy category of U-complexes is an additive category. In this paper, we propose a weakly chain U-complex by changing the second condition of the chain U-complex. We prove that the homotopy category ofweakly U-complexes is a triangulated category.

Core and dual core inverses of a sum of morphisms

Let C be an additive category with an involution *. Suppose that ? : X ? X is a morphism of C with core inverse ?# : X ? X and ? : X ? X is a morphism of C such that 1X + ?#? is invertible. Let ? = (1X+?#?)-1, ? = (1X+??#)-1, ? = (1X-??#)??(1X-?#?), ? = ?(1X-?#?)?-1??#?,? = ??#??-1(1X-??#)?,? = ?*(?#(*?*(1X-??#)?. Then f = ? + ? ? ? has a core inverse if and only if 1X-?, 1X-? and 1X-? are invertible. Moreover, the expression of the core inverse of f is presented. Let R be a unital *-ring and J(R) its Jacobson radical, if a ? R# with core inverse a # and j ? J(R), then a + j ? R# if and only if (1-aa#)j(1+a#j)-1(1-a#a) = 0. We also give the similar results for the dual core inverse.

ON COMPONENTS OF STABLE AUSLANDER–REITEN QUIVERS THAT CONTAIN HELLER LATTICES: THE CASE OF TRUNCATED POLYNOMIAL RINGS

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$, and we consider the additive category consisting of $A$-lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$-module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$. Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

Topological rigidity problems

We survey the recent results and current issues on the topological rigidity problem for closed aspherical manifolds, i.e., connected closed manifolds whose universal coverings are contractible. A number of open problems and conjectures are presented during the course of the discussion. We also review the status and applications of the Farrell-Jones Conjecture for algebraic \(K\)-and \(L\)-theory for a group ring $RG$ and coefficients in an additive category. These conjectures imply many other well-known and important conjectures. Examples are the Borel Conjecture about the topological rigidity of closed aspherical manifolds, the Novikov Conjecture about the homotopy invariance of higher signatures and the Conjecture for vanishing of the Whitehead group. We here present the status of the Borel, Novikov and vanishing of the Whitehead group Conjectures.

NON-COMMUTATIVE LOCALIZATIONS OF ADDITIVE CATEGORIES AND WEIGHT STRUCTURES

In this paper we demonstrate thatnon-commutative localizationsof arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category$\text{}\underline{C}$by a set$S$of morphisms in the heart$\text{}\underline{Hw}$of a weight structure$w$on it one obtains a triangulated category endowed with a weight structure$w^{\prime }$. The heart of$w^{\prime }$is a certain version of the Karoubi envelope of the non-commutative localization$\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$(of$\text{}\underline{Hw}$by$S$). The functor$\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of$S$invertible. For any additive category$\text{}\underline{A}$, taking$\text{}\underline{C}=K^{b}(\text{}\underline{A})$we obtain a very efficient tool for computing$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$coincides with the ‘abstract’ localization$\text{}\underline{A}[S^{-1}]$(as constructed by Gabriel and Zisman) if$S$contains all identity morphisms of$\text{}\underline{A}$and is closed with respect to direct sums. We apply our results to certain categories of birational motives$DM_{gm}^{o}(U)$(generalizing those defined by Kahn and Sujatha). We define$DM_{gm}^{o}(U)$for an arbitrary$U$as a certain localization of$K^{b}(Cor(U))$and obtain a weight structure for it. When$U$is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general$U$the result is completely new. The existence of the correspondingadjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over$U$.