Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories14610175510.52547/cgasa.2021.101755ENKaique Matias De AndradeRobertoInstituto de Matemática e Estatística, Universidade de Sao Paulo, Brazil.0000-0001-7136-8951Hugo LuizMarianoInstituto de Matemática e Estatística, Universidade de Sao Paulo, Brazil.0000-0002-9745-2411Journal Article20211127We build on previous work on multirings ([17]) that provides<br />generalizations of the available abstract quadratic forms theories (special<br />groups and real semigroups) to the context of multirings ([10], [14]). Here<br />we raise one step in this generalization, introducing the concept of pre-special<br />hyperfields and expand a fundamental tool in quadratic forms theory to the<br />more general multivalued setting: the K-theory. We introduce and develop<br />the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor’s<br />K-theory ([11]) and Special Groups K-theory, developed by Dickmann-<br />Miraglia ([5]). We develop some properties of this generalized K-theory, that<br />can be seen as a free inductive graded ring, a concept introduced in [2] in<br />order to provide a solution of Marshall’s Signature Conjecture.https://cgasa.sbu.ac.ir/article_101755_c8a4e43e111cf70f030b33574d574259.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701Expanding Belnap 2: the dual category in depth478410244310.52547/cgasa.2022.102443ENAndrew P. K.CraigDepartment of Mathematics
and Applied Mathematics
University of Johannesburg
PO Box 524, Auckland Park, 2006,
South AfricaBrian A.DaveyDepartment of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia.orcid.org/0000-0002-1200-1989MiroslavHaviarDepartment of Mathematics
Faculty of Natural Sciences,
M. Bel University
Tajovskeho 40, 974~01 Banska Bystrica, Slovakia.Journal Article20220616Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled How a computer should think. Prioritised default bilattices include not only Belnap’s four values, for ‘true’ (t), ‘false’(f), ‘contradiction’(⊤) and ‘no information’ (⊥), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, Jn, for n ⩾ 0, with J0 being Belnap’s seminal example. We gave a duality for the variety Vn generated by Jn, with the dual category Xn consisting of multi-sorted topological structures. Here we study the dual category in depth. We axiomatise the category Xn and show that it is isomorphic to a category Yn of single-sorted topological structures. The objects of Yn are ranked Priestley spaces endowed with a continuous retraction. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in Vn via its dual in Yn; as an application we show that the size of the free algebra FVn(1) is given by a polynomial in n of degree 6.https://cgasa.sbu.ac.ir/article_102443_eee83c99c3f1c6a0c84198810233eae9.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701On some properties of the space of minimal prime ideals of 𝐶𝑐 (𝑋)8510010262210.52547/cgasa.2022.102622ENZahraKeshtkarDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.RostamMohamadianDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.MehrdadNamdariDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.MaryamZeinaliDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.Journal Article20220808In this article we consider some relations between the topological properties of the spaces X and Min(C<sub>c</sub> (X)) with algebraic properties of C<sub>c</sub> (X). We observe that the compactness of Min(C<sub>c</sub> (X)) is equivalent to the von-Neumann regularity of q<sub>c</sub> (X), the classical ring of quotients of C<sub>c</sub> (X). Furthermore, we show that if 𝑋 is a strongly zero-dimensional space, then each contraction of a minimal prime ideal of 𝐶(𝑋) is a minimal prime ideal of C<sub>c</sub>(X) and in this case 𝑀𝑖𝑛(𝐶(𝑋)) and Min(C<sub>c</sub> (X)) are homeomorphic spaces. We also observe that if 𝑋 is an F<sub>c</sub>-space, then Min(C<sub>c</sub> (X)) is compact if and only if 𝑋 is countably basically disconnected if and only if Min(C<sub>c</sub>(X)) is homeomorphic with β<sub>0</sub>X. Finally, by introducing z<sup>o</sup><sub>c</sub>-ideals, countably cozero complemented spaces, we obtain some conditions on X for which Min(C<sub>c</sub> (X)) becomes compact, basically disconnected and extremally disconnected.https://cgasa.sbu.ac.ir/article_102622_fabfade2e239fe905af15ccfebc0a21e.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701Universal extensions of specialization semilattices10111610246710.52547/cgasa.2022.102467ENPaoloLippariniDipartimento di Matematica, Viale della Ricerca Scientifica Non Chiusa, Universit`a di Roma “Tor Vergata”, I-00133 Rome, Italy.Journal Article20220622A specialization semilattice is a join semilattice together with a coarser preorder ⊑ satisfying an appropriate compatibility condition. If X is a topological space, then (P(X),∪,⊑) is a specialization semilattice, where x ⊑ y if x ⊆ Ky, for x, y ⊆ X, and K is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. Here we describe the universal embedding of a specialization semilattice into an additive closure semilattice.https://cgasa.sbu.ac.ir/article_102467_22fa793c505863fa9d7697bafd46728e.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701Coverings and liftings of generalized crossed modules11714010249110.52547/cgasa.2022.102491ENGamzeAytekin ArıcıDepartment of Mathematics, Aksaray University, P. O. Box 68100, Aksaray, Turkey.0000-0002-3412-3856TunçarŞahanDepartment of Mathematics, Aksaray University, P. O. Box 68100, Aksaray, Turkey.0000-0002-6552-4695Journal Article20220628In the theory of crossed modules, considering arbitrary selfactions instead of conjugation allows for the extension of the concept of crossed modules and thus the notion of generalized crossed module emerges. In this paper we give a precise definition for generalized cat<sup>1</sup>-groups and obtain a functor from the category of generalized cat<sup>1</sup>-groups to generalized crossed modules. Further, we introduce the notions of coverings and liftings for generalized crossed modules and investigate properties of these structures. Main objective of this study is to obtain an equivalence between the category of coverings and the category of liftings of a given generalized crossed module (A,B, α).https://cgasa.sbu.ac.ir/article_102491_f8d8b6da716db7ffdad198c9457e4b61.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701On nominal sets with support-preorder14117210262310.52547/cgasa.2022.102623ENAliyehHossinabadiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.MahdiehHaddadiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.KhadijehKeshvardoostDepartment of Mathematics, Velayat University, Iranshahr, Sistan and
Balouchistan, Iran.Journal Article20220808Each nominal set 𝑋 can be equipped with a preorder relation ⪯ defined by the notion of support, so-called support-preorder. This preorder also leads us to the support topology on each nominal set. We study support-preordered nominal sets and some of their categorical properties in this paper. We also examine the topological properties of support topology, in particular separation axioms.https://cgasa.sbu.ac.ir/article_102623_81d5fc73f55777aac4d773bc2d38fde1.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701On injective objects and existence of injective hulls in 𝑄-TOP/(𝑌, 𝜎)17320210264010.52547/cgasa.2022.102640ENHarshitaTiwariDepartment of Mathematical Sciences, Indian Institute of Technology,
Banaras Hindu University, Varanasi-221005, India.RekhaSrivastavaDepartment of Mathematical Sciences, Indian Institute of Technology,
Banaras Hindu University, Varanasi-221005, India.Journal Article20220813In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class of embeddings in the category 𝑄-<strong>TOP</strong> of 𝑄-topological spaces) in the comma category 𝑄-<strong>TOP</strong>/(𝑌,𝜎), when (𝑌,𝜎) is a stratified 𝑄-topological space, with the help of their 𝑇<sub>0</sub>-reflection. Further, we have proved that for any 𝑄-topological space (𝑌,𝜎), the existence of an injective hull of ((𝑋, 𝜏), 𝑓 ) in the comma category 𝑄-<strong>TOP</strong>/(𝑌, 𝜎) is equivalent to the existence of an injective hull of its 𝑇<sub>0</sub>-reflection ((𝑋 ̃,𝜏 ̃), 𝑓 ̃) in the comma category Q-<strong>TOP</strong>/(𝑌 ̃, 𝜎 ̃ ) (and in the comma category 𝑄-<strong>TOP</strong><sub>0</sub>/(𝑌 ̃, 𝜎 ̃ ), where 𝑄-<strong>TOP</strong><sub>0</sub> denotes the category of 𝑇<sub>0</sub>-𝑄-topological spaces).https://cgasa.sbu.ac.ir/article_102640_5e972639559d03b0396400e61c365bb7.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585317120220701Quantum determinants in ribbon category20323210262110.52547/cgasa.2022.102621ENHananChoulliMathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of
Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.KhalidDraouiMathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of
Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.HakimaMouanisMathematical Sciences and Applications Laboratory, Department of Mathematics, Faculty of
Sciences Dhar Al Mahraz, P. O. Box 1796, University Sidi Mohamed Ben Abdellah Fez, Morocco.Journal Article20220808The aim of this paper is to introduce an abstract notion of determinant which we call quantum determinant, verifying the properties of the classical one. We introduce R−basis and R−solution on rigid objects of a monoidal 𝐴𝑏−category, for a compatibility relation R, such that we require the notion of duality introduced by Joyal and Street, the notion given by Yetter and Freyd and the classical one, then we show that R−solutions over a semisimple ribbon 𝐴𝑏−category form as well a semisimple ribbon 𝐴𝑏−category. This allows us to define a concept of so-called quantum determinant in ribbon category. Moreover, we establish relations between these and the classical determinants. Some properties of the quantum determinants are exhibited.https://cgasa.sbu.ac.ir/article_102621_705fc5756a175ca8150b223412b93f27.pdf