Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Inductive graded rings, hyperfields and quadratic forms10469610.48308/cgasa.2024.232583.1417ENKaique Matias De AndradeRobertoInstitute of Mathematics and Statistics, University of São Paulo, Brazil.0000-0001-7136-8951Hugo LuizMarianoInstitute of Mathematics and Statistics, University of São Paulo, Brazil.0000-0002-9745-2411Journal Article20230804In [6] we developed a k-theory for the category of hyperbolic hyperfields (a category that contains a copy of the category of (pre)special groups): this construction extends, simultaneously, Milnor's k-theory ([20]) and Dickmann-Miraglia's k-theory ([13]). An abstract environment that encapsulate all them, and of course, provide an axiomatic approach to guide new extensions of the concept of K-theory in the context of the algebraic and abstract theories of quadratic forms is given by the concept of inductive graded rings a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in realm the algebraic theory of quadratic forms for Pythagorean fields. The goal of this work is twofold: (i) to provide a detailed analysis of some categories of inductive graded ring - a concept introduced in [9] in order to provide a solution of Marshall's signature conjecture in the algebraic theory of quadratic forms; (ii) apply this analysis to deepen the connections between the category of special hyperfields ([6]) - equivalent to the category of special groups ([10]) and the categories of inductive graded rings.https://cgasa.sbu.ac.ir/article_104696_d7ea43f526970c0dcc1908cd9d77679e.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Characterizing cogenerating and finitely cogenerated $S$-acts10645810.48308/cgasa.2024.235961.1497ENRoghaiehKhosraviDepartment of Mathematics, Faculty of Sciences, Fasa University, Fasa, Iran.0000-0003-0583-3936XingliangLiangDepartment of Mathematics, Shaanxi University of Science and Technology, Xi'an, Shaanxi, P.R. China.0000-0002-4517-8684MohmmadRoueentanCollege of Engineering, Lamerd Higher Education Center, Shiraz University of Technology, Lamerd, Iran.0000-0001-8755-6850Journal Article20240607In this paper, we introduce cogenerating classes of $S$-acts as those that can be used to cogenerate $S$-acts in an appropriate sense. Next, finitely cogenerated $S$-acts are characterized by the property that their socle is finitely cogenerated and large in the $S$-act. Further, we investigate the $S$-acts cogenerating $S_S$, or generating the injective envelope $E(S)$ of $S_S$. This leads us to introduce the classes of cofaithful and subgenerator $S$-acts as the dual notions of faithful $S$-acts, which lie strictly between the classes of generator and faithful $S$-acts. Ultimately, we study relations between the cogenerating classes, finitely cogenerated $S$-acts, and the recently introduced new classes of $S$-acts.https://cgasa.sbu.ac.ir/article_106458_a0876bd594362e51534eda0537faf697.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Picard group of dual categories10491510.48308/cgasa.2024.234639.1470ENAdrianaMejía CastañoDepartment of Mathematics and Statistics, Universidad del Norte, Barranquilla, Colombia.0000-0003-4486-9165Journal Article20240130 We provide an explicit description of the Picard group (the group of isomorphism classes of invertible objects, those that have an inverse under the tensor product) of the dual category of the category of comodules over a supergroup algebra, by using the description of this group for group-theoretical categories. In fact we prove that there is a subgroup relation between these groups. As an interest application of this group in a modular context, it can be used to construct examples of symmetric special Frobenius algebras. They also plays an important role in the theory of braided tensorcategories for the classification of group extensions of fusion categories.https://cgasa.sbu.ac.ir/article_104915_f4874f4fb472d4722b519d08e1325589.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Classical prime subhypermodules and related extensions10489710.48308/cgasa.2024.234345.1461ENMahdiAnbarloeiFaculty of Sciences , Department of Mathematics, Imam Khomeini International University, Qazvin, Iran.0000-0003-3260-2316Journal Article20240102In this paper, we extend the notion of prime subhypermodules<br />to $n$-ary classical prime, $n$-ary weakly classical prime and $n$-ary $\phi$-classical prime subhypermodules of an $(m,n)$-hypermodule over a commutative Krasner $(m,n)$-hyperring. Many properties and characterizations of them are introduced. Moreover, we investigate the behavior of these structures under hypermodule homomorphisms, quotient hypermodules and cartesian product. We think the knowledge gained in this setting provides a significant step in the general investigation of subhypermodules. https://cgasa.sbu.ac.ir/article_104897_78f607e7e2c390ccf2b135ed399455de.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Slicing points in a pointfree adjunction for $T_D$ partial spaces10500810.48308/cgasa.2024.235898.1494ENJohnFrithDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.0000-0001-6421-3177AnnelieseSchauerteDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag Rondebosch, 7701, South Africa.0000-0002-8540-3997Journal Article20240601The $T_D$ axiom, a low order separation axiom between $T_0$ and $T_1$, has been of interest to classical topologists for some time; latterly it has also proved interesting to pointfree topologists. Here we investigate it in the context of partial spaces and partial frames (think: $\sigma$-frames, $\kappa$-frames, frames, bounded distributive lattices). We establish an adjunction between the category of $T_D$ partial spaces with continuous maps and the category of partial frames with $D$-homomorphisms. Several standard tools (covered primes, right adjoints, point closures) are not appropriate in our setting; we use linked pairs and slicing points instead. Of particular interest are the slicing points of free frames and congruence frames.We examine the fixed objects of the adjunction; both similarities and differences to the classical situation become clear. In particular, there are compact Hausdorff partial spaces that are not $T_D$. We introduce sharp partial frames, those for which all points are slicing and characterize these as well as the $T_D$ spatial and strongly $T_D$ spatial partial frames. We conclude with a comparison of sober and $T_D$ partial spaces.https://cgasa.sbu.ac.ir/article_105008_c64eca8a55f9ea302cf419ce51b14e6b.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701On homological classification of monoids by Condition $\mathbf{(P_{sc})}$ and new classification on Condition $\mathbf{(P_{E})}$10491610.48308/cgasa.2024.234899.1472ENHosseinMohammadzadeh SaanyDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, IranMortezaJafariDepartment of Mathematics Education, Farhangian University, Tehran, Iran.LeilaNouriDepartment of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran0000-0002-2240-583XJournal Article20240223In 1997, Golchin and Renshaw introduced Condition $(P_E)$ and showed that this condition implies weak flatness, although the converse is not generally valid.<br />In this paper, we present Condition $(P_{sc})$ as a generalization of Condition $(P_E)$. We also see that Condition $(P_{sc})$ implies weak flatness, but the converse is not necessarily true. However, for left $PSF$ monoids the converse is holds. Moreover, we discuss some general properties and provide a homological classification of monoids by comparing Condition $(P_{sc})$ with some other properties.<br />Furthermore, a new homological classification of monoids is presented by comparing Condition $(P_E)$ with other properties.https://cgasa.sbu.ac.ir/article_104916_ce7c8b6b17fc2673faf988d0b7166feb.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Ideals and congruences in $L$-algebras and pre-$L$-algebras10502710.48308/cgasa.2024.232671.1419ENMarinoGranInstitut de Recherche en Math\'{e}matique et Physique, Universit\'{e} catholique de Louvain, 1348 Louvain-la-Neuve, Belgique.0000-0002-9859-5238AlbertoFacchiniDipartimento di Matematica ``Tullio Levi-Civita'', Universit\`a di Padova, 35121 Padova, Italy.MaraPompiliDepartment of Mathematics and Scientific Computing, University of Graz, 8010 Graz, Austria0000-0003-0681-7241Journal Article20230808 We link the recent theory of $L$-algebras to previous notions of Universal Algebra and Categorical Algebra concerning subtractive varieties, commutators, multiplicative lattices, and their spectra. We show that the category of $L$-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of $L$-algebras nor that of pre-$L$-algebras are Mal'tsev categories, hence in particular they are not semi-abelian. Therefore $L$-algebras are a rather peculiar example of an algebraic structure.https://cgasa.sbu.ac.ir/article_105027_79e662f20445aae59c27ef1beb8ac083.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585323120250701Baer criterion in locally presentable categories10467410.48308/cgasa.2024.235938.1495ENMojganMahmoudiMojgan Mahmoudi; Professor
Department of Mathematics,
Shahid Beheshti University,0000-0002-7556-8536AlirezaMehdizadehFaculty of Mathematical Sciences, Department of Mathematics, Shahid Beheshti University, 19839 Tehran, Iran.0009-0003-3555-0766Journal Article20240605In this paper, some Baer type criteria are considered for locally presentable categories. Recalling the notion of the classical Baer criterion for injectivity, it is shown that a locally presentable category which has enough injectives and coproduct injections, which are monomorphisms, satisfy such criterion if and only if the class of its injective objects is accessibly embedded in the category. Also, it is shown that this criterion is equivalent to the Baer type criterion that injectivity is equivalent to injectivity with respect to a subclass of monomorphisms.<br /><br />It is also proved some Baer type criteria for $\lambda$-presentable categories for injectivity with respect to monomorphisms with $\lambda$-presentable domains and codomains, for a regular cardinal number $\lambda$.<br />In particular, some Baer type criteria is found for varieties.https://cgasa.sbu.ac.ir/article_104674_f23ee2c88e5ad2bdbc5cf3476c8dd7eb.pdf