Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201Formal balls of Q-categories11810291410.52547/cgasa.2022.102914ENXianbo YangSchool of Mathematics, Sichuan University, Chengdu, ChinaDexue ZhangSchool of Mathematics, Sichuan University, Chengdu, ChinaJournal Article20221202The construction of the formal ball model for metric spaces due to Edalat and Heckmann was generalized to Q-categories by Kostanek and Waszkiewicz, where Q is a commutative and unital quantale. This paper concerns the influence of the structure of the quantale Q on the connection between Yoneda completeness of Q-categories and directed completeness of their sets of formal balls. In the case that Q is the unit interval [0, 1] equipped with a continuous t-norm &, it is shown that in order that Yoneda completeness of each Q-category be equivalent to directed completeness of its set of formal balls, a necessary and sufficient condition is that the t-norm & is Archimedean.https://cgasa.sbu.ac.ir/article_102914_351656fec0984f81ec40c98cc66412ed.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201On saturated prefilter monads194110293410.52547/cgasa.2022.102934ENGao ZhangInstitute of Mathematics, Nanjing Normal University, Nanjing, ChinaWei HeInstitute of Mathematics, Nanjing Normal UniversityJournal Article20221213In this paper we show that the prime saturated prefilter monads are sup-dense and interpolating in saturated prefilter monads. It follows that CNS spaces are the lax algebras for prime saturated prefilter monads. As for the algebraic part, we prove that the Eilenberg-Moore algebras for saturated prefilter monads are exactly continuous I-lattices.https://cgasa.sbu.ac.ir/article_102934_2e33aea58b98b9d914eb6ead986f84a8.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201Action graph of a semigroup act & its functorial connection438010301910.52547/cgasa.2022.103019ENPromit MukherjeeMathematics, Faculty of Science, Jadavpur University, Kolkata-700032
IndiaRajlaxmi MukherjeeDepartment of Mathematics, Garhbeta College, Paschim Medinipur-721127, IndiaSujit KumarSardarMathematics, Faculty of Science, Jadavpur University, Kolkata-700032, India.0000-0001-7837-0835Journal Article20221228In this paper we define C-induced action graph G(S,a,C;A) corresponding to a semigroup act (S,a,A) and a subset C of S. This generalizes many interesting graphs including Cayley Graph of groups and semigroups, Transformation Graphs (TRAG), Group Action Graphs (GAG), Derangement Action Graphs, Directed Power Graphs of Semigroups etc. We focus on the case when C = S and name the digraph, so obtained, as Action Graph of a Semigroup Act (S, a, A). Some basic structural properties of this graph follow from algebraic properties of the underlying semigroup and its action on the set. Action graph of a strongly faithful act is also studied and graph theoretic characterization of a strongly faithful semigroup act as well as that of idempotents in a semigroup are obtained. We introduce the notion of strongly transitive digraphs and based on this we characterize action graphs of semigroup acts in the class of simple digraphs. The simple fact that morphism between semigroup acts leads to digraph homomorphism between corresponding action graphs, motivates us to represent action graph construction as a functor from the category of semigroup acts to the category of certain digraphs. We capture its functorial properties, some of which signify previous results in terms of Category Theory.https://cgasa.sbu.ac.ir/article_103019_fdac39742402364cd4b8caffe3abefc0.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201On (semi)topology L-algebras8110310312110.52547/cgasa.2023.103121ENMona Aaly KologaniDepartment of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran0000-0002-5234-2876Journal Article20230131Here, we define (semi)topological L-algebras and some related results are approved. Then we deduce conditions that mention an L-algebra to be a semi-topological or a topological L-algebra and we check some attributes of them. Chiefly, we display in an L-algebra L, if (L, ↠, τ ) is a semi-topological L-algebra and {1} is an open set or L is bounded and satisfies the double negation property, then (L,τ) is a topological L-algebra. Finally, we construct a discrete topology on a quotient L-algebra, under suit- able conditions. Also, different kinds of topology such as T<sub>0</sub> and Hausdorff are investigated.https://cgasa.sbu.ac.ir/article_103121_e423455a15e8af77185a5171e0f66417.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201Classification of 1-absorbing comultiplication modules over a pullback ring10512910314310.52547/cgasa.2023.103143ENFarkhondeh FarzalipourDepartment of Mathematics, Payame Noor University, P.O.BOX 19395-3697, Tehran, Iran.0000-0003-2494-5466Peyman GhiasvandDepartment of Mathematics, Payame Noor University, P.O.BOX 19395- 3697, Tehran, Iran.0000-0003-4084-7057Journal Article20230202One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.https://cgasa.sbu.ac.ir/article_103143_62a3f40ad010bb5caa7ee3ac609a973d.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201The prime state ideal theorem in state residuated lattices13115310328810.52547/cgasa.2023.103288ENFrancis WoumfoDepartement of Mathematics and Computer Science, Faculty of Sciences, University of Dschang, Dschang, Cameroon0000-0002-0479-4309Etienne Romuald AlomoTemgouaDepartment of Mathematics, Ecole Normale Sup´erieure, de Yaound´e, University of Yaound´e 1, P.O. Box 47, Yaound´e, Cameroon.0000-0001-6338-090XCelestin LeleDepartment of Mathematics and Computer Science, Faculty of Science, University of Dschang, Dschang, Cameroon0000-0001-6338-090XJournal Article20230313The aim of this paper is to establish the prime state ideal theorem in state residuated lattices (SRLs). We study the state ideals lattice $\mathcal{SI}(L)$ of a state residuated lattice $(L, \varphi)$ and prove that it is a complete Brouwerian lattice in which the meet and the join of any two compact elements are compact (coherent frame). We characterize the notion of prime state ideals in SRLs. In addition, we establish the condition for which the lattice $\mathcal{SI}(L)$ is a Boolean algebra. https://cgasa.sbu.ac.ir/article_103288_7c80a9cedf8e1ba5002830b5d68d31cf.pdfShahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585318120230201Internal Neighbourhood Structures II: Closure and closed morphisms15522310328910.52547/cgasa.2023.103289ENPartha Pratim GhoshDepartment of Mathematical Sciences, University of South Africa, Unisa Science Campus, corner of Christian de Wet \& Pioneer Avenue, Florida 1709, Johannesburg, Gauteng, South Afric, National Institute for Theoretical and Computational Sciences (NITheCS), South Africa.0000-0001-5519-7056Journal Article20220813 Internal preneighbourhood spaces inside any finitely complete category with finite coproducts and proper factorisation structure were first introduced in \cite{2020}. This paper proposes a closure operation on internal preneighbourhood spaces and investigates closed morphisms and its close allies. Consequently it introduces analogues of several well-known classes of topological spaces for preneighbourhood spaces. Some preliminary properties of these spaces are established in this paper. The results of this paper exhibit that preneighbourhood systems are more general than closure operators and conveniently allows identifying properties of classes of morphisms independent of \emph{continuity} of morphisms with respect to induced closure operators.https://cgasa.sbu.ac.ir/article_103289_fae708a78b452995e92ce69b61e4a36d.pdf