Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801Cancel culture12710358010.48308/cgasa.19.1.1ENJonathan DavidFarleyDepartment of Mathematics, Morgan State University, Baltimore, MD 21251, United States of AmericaJournal Article20230610Let A, B, C, and D be posets. Assume C and D are finite with a greatest element. Also assume that A<sup>C</sup> ≅B <sup>D</sup>. Then there exist posets E, X, Y , and Z such that A ≅E <sup>X</sup>, B ≅E <sup>Y</sup> , C≅Y ×Z, and D≅X×Z. If C≅D, then A≅B. This generalizes a theorem of Jónsson and McKenzie, who proved it when A and B were meet-semilattices.https://cgasa.sbu.ac.ir/article_103580_7e019629f987fc8498db5f7b71403816.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801Composition series on (Rees) congruences of S-acts.294610337510.48308/cgasa.19.1.29ENRoghaieh KhosraviDepartment of Mathematics, Faculty of Sciences, Fasa University, Fasa, Iran.Mohammad RoueentanCollege of Engineering, Lamerd Higher Education Center, Lamerd, Iran.Journal Article20230413In this paper, we study composition series of subacts or congruences of S-acts. It is shown that composition series of subacts are exactly those that are both Rees artinian and Rees noetherian, i.e. those satisfying both ascending and descending chain conditions on subacts. But this is not valid for the case of composition series of congruences in general. We prove that the properties of having composition series of subacts or congruences are inherited in Rees short exact sequences. Also, we discuss whenever two composition series of subacts or congruences have the same length and they are equivalent.https://cgasa.sbu.ac.ir/article_103375_d0c9c4f6c2a778dc71a79da62ac92bc1.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801Determinant and rank functions in semisimple pivotal Ab-categories478010382910.48308/cgasa.19.1.47ENKhalid DraouiMathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.0000-0001-9879-4096Hanan ChoulliMathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.0000-0002-7260-882XHakima MouanisMathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.0000-0002-9654-8139Journal Article20230505We investigate and generalize quantum determinants to semisimple spherical and pivotal categories. It is well known that traces are preserved by strong tensor functors; we show on one hand that in fact, weaker conditions on a functor are sufficient to continue preserving traces. On the other hand, we prove that these determinants are well-behaved under strong tensor functors. Further, we introduce a notion of domination rank for objects of a semisimple pivotal category and prove similar properties of the ordinary case. Furthermore, we expand the determinantal and McCoy ranks to introduce a morphism quantum rank function on a semisimple pivotal category.https://cgasa.sbu.ac.ir/article_103829_ff81f70abc504501a6a6837d564fda92.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801On free acts over semigroups and their lattices of radical subacts8110210373610.48308/cgasa.19.1.81ENMohammad Ali NaghipoorDepartment of Mathematics, Jahrom University, Jahrom, Iran.0000-0002-2453-9604Journal Article20230512This study aims to investigate free objects in the category of acts over an arbitrary semigroup S. We consider two generalizations of free acts over arbitrary semigroups, namely acts with conditions (F1) and (F2), and give some new results about (minimal) prime subacts and radical subacts of any S-act with condition (F1). Furthermore, some lattice structures for some collections of radical subacts of free S-acts are introduced. We also obtain some results about the relationship between radical subacts of free S-acts and radical ideals of S. Moreover, for any prime ideal P of a semigroup S with a zero, we find a one-to-one correspondence between the collections of P-prime subacts of any two free S-acts. Also, it is shown that all free S-acts have isomorphic lattices of radical multiplication subacts.https://cgasa.sbu.ac.ir/article_103736_312f4bc60914a9c197921bed98249f59.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801Morita equivalence of certain crossed products10312510384210.48308/cgasa.19.1.103ENAdriana Mejia CastañoDepartament of Matemathics and statistics, Universidad del Norte, Colombia0000-0003-4486-9165Journal Article20230718We introduce an alternative criterion for Morita equivalence over graded tensor categories using equivariant centers and equivariantizations. While Morita equivalence has been extensively studied in the context of fusion categories, primarily through the examination of their centers, recent advancements have broadened its scope to encompass graded tensor categories. This paper presents a novel criterion for characterizing Morita equivalence in graded tensor categories by leveraging equivariant centers and equivariantizations. Notably, the identification of Morita equivalence can be expedited when the Brauer Picard groups are known, offering an efficient approach to establishing the equivalence relationship.<br />To generalize the properties of fusion categories to finite tensor categories, we utilize theconcept of an exact module category, which was introduced by Etingof and Ostrik. Exact module categories offer an intermediary restriction between the semisimplemodule categories of a fusion category and more general cases that may not be semisimpleor finite.https://cgasa.sbu.ac.ir/article_103842_0f01fb79826ad2b95bc8f9a86316c46f.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801On C-injective generalized hyper S-acts12714110388110.48308/cgasa.19.1.127ENMaedeh GhasempourDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranHamid RasouliDepartment of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, IranAli IranmaneshDepartment of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, IranHasan BarzegarDepartment of Mathematics, Tafresh University, Tafresh 39518-
79611, IranAbolfazl TehranianDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranJournal Article20230825This paper explores generalized hyper S-acts (GHS-acts) over a hypermonoid S as generalizations of monoid acts within the context of algebraic hyperstructures. Specifically, we extend the definition of C-injectivity to GHS-acts and investigate their internal and homological properties. It is established that for being GHS-injectivity of GHS-acts with a fixed element, it suffices to consider all<br />inclusions from cyclic GHS-subacts into indecomposable ones. Then we introduce<br />new concepts known as semi-injectivity and semi-C-injectivity. By providing examples, we demonstrate that injectivity and semi-injectivity (C-injectivity and semiC-injectivity) are different concepts for GHS-acts, whereas they are the same in the context of acts over monoids. It is also shown that all pure GHS-acts are injective if and only if all pure cyclic GHS-acts are C-injective. Furthermore, we establish an equivalent condition on a hypermonoid S such that all quotients of SS exhibit semi-injectivity. Finally, we derive an equivalent condition for a hypermonoid to be<br />classified as semi-injective.https://cgasa.sbu.ac.ir/article_103881_92769525788a374436c796f5305133ba.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801Green's relations on ordered n-ary semihypergroups14316710385810.48308/cgasa.19.1.143ENJukkrit DaengsaenDepartment of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand.Sorasak LeeratanavaleeDepartment of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand.0000-0001-8818-6134Journal Article20230421In this paper, we introduce the concept of weak $i$-hyperfilters of ordered $n$-ary<br />semihypergroups, where a positive integer 1 ≤ i ≤ n and n ≥ 2, and then discuss their<br />related properties. We define the Green’s relations $M_i, J , H$ and $K$ on ordered $n$-ary<br />semihypergroups and investigate the relationships between the Green’s relations and the<br />equivalence relation $W_i$, which is generated by the weak $i$-hyperfilters. Also, we give<br />the characterizations of intra-regular ordered $n$-ary semihypergroups via the properties<br />of weak $i$-hyperfilters. Finally, we introduce the concepts of $(i-, Λ-)$duo ordered $n$-ary<br />semihypergroups and establish some interesting properties.https://cgasa.sbu.ac.ir/article_103858_f6acfd659d49b01664cafc06a330d3a9.pdf