Categories and General Algebraic Structures with ApplicationsCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
Fri, 24 May 2019 05:59:55 +0100FeedCreatorCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
Feed provided by Categories and General Algebraic Structures with Applications. Click to visit.Cover for Vol. 10, No. 1.
http://cgasa.sbu.ac.ir/article_81554_10418.html
Mon, 31 Dec 2018 20:30:00 +0100An equivalence functor between local vector lattices and vector lattices
http://cgasa.sbu.ac.ir/article_61405_10418.html
We call a local vector lattice any vector lattice with a distinguished positive strong unit and having exactly one maximal ideal (its radical). We provide a short study of local vector lattices. In this regards, some characterizations of local vector lattices are given. For instance, we prove that a vector lattice with a distinguished strong unit is local if and only if it is clean with non no-trivial components. Nevertheless, our main purpose is to prove, via what we call the radical functor, that the category of all vector lattices and lattice homomorphisms is equivalent to the category of local vectors lattices and unital (i.e., unit preserving) lattice homomorphisms.Mon, 30 Apr 2018 19:30:00 +0100State filters in state residuated lattices
http://cgasa.sbu.ac.ir/article_57443_10418.html
In this paper, we introduce the notions of prime state filters, obstinate state filters, and primary state filters in state residuated lattices and study some properties of them. Several characterizations of these state filters are given and the prime state filter theorem is proved. In addition, we investigate the relations between them.Mon, 31 Dec 2018 20:30:00 +0100Lattice of compactifications of a topological group
http://cgasa.sbu.ac.ir/article_61406_10418.html
We show that the lattice of compactifications of a topological group $G$ is a complete lattice which is isomorphic to the lattice of all closed normal subgroups of the Bohr compactification $bG$ of $G$. The correspondence defines a contravariant functor from the category of topological groups to the category of complete lattices. Some properties of the compactification lattice of a topological group are obtained.Mon, 31 Dec 2018 20:30:00 +0100On the property $U$-($G$-$PWP$) of acts
http://cgasa.sbu.ac.ir/article_50746_10418.html
In this paper first of all we introduce Property $U$-($G$-$PWP$) of acts, which is an extension of Condition $(G$-$PWP)$ and give some general properties. Then we give a characterization of monoids when this property of acts implies some others. Also we show that the strong (faithfulness, $P$-cyclicity) and ($P$-)regularity of acts imply the property $U$-($G$-$PWP$). Finally, we give a necessary and sufficient condition under which all (cyclic, finitely generated) right acts or all (strongly, $Re$-) torsion free (cyclic, finitely generated) right acts satisfy Property $U$-($G$-$PWP$).Mon, 31 Dec 2018 20:30:00 +0100A Universal Investigation of $n$-representations of $n$-quivers
http://cgasa.sbu.ac.ir/article_63576_10418.html
noindent We have two goals in this paper. First, we investigate and construct cofree coalgebras over $n$-representations of quivers, limits and colimits of $n$-representations of quivers, and limits and colimits of coalgebras in the monoidal categories of $n$-representations of quivers. Second, for any given quivers $mathit{Q}_1$,$mathit{Q}_2$,..., $mathit{Q}_n$, we construct a new quiver $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$, called an $n$-quiver, and identify each category $Rep_k(mathit{Q}_j)$ of representations of a quiver $mathit{Q}_j$ as a full subcategory of the category $Rep_k(mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}})$ of representations of $mathscr{Q}_{!_{(mathit{Q}_1, mathit{Q}_2,..., mathit{Q}_n)}}$ for every $j in {1,2,ldots , n}$.Mon, 31 Dec 2018 20:30:00 +0100Mappings to Realcompactifications
http://cgasa.sbu.ac.ir/article_61474_10418.html
In this paper, we introduce and study a mapping from the collection of all intermediate rings of $C(X)$ to the collection of all realcompactifications of $X$ contained in $beta X$. By establishing the relations between this mapping and its converse, we give a different approach to the main statements of De et. al. Using these, we provide different answers to the four basic questions raised in Acharyya et.al. Finally, we give some notes on the realcompactifications generated by ideals.Mon, 31 Dec 2018 20:30:00 +0100Applications of the Kleisli and Eilenberg-Moore 2-adjunctions
http://cgasa.sbu.ac.ir/article_76725_10418.html
In 2010, J. Climent Vidal and J. Soliveres Tur developed, among other things, a pair of 2-adjunctions between the 2-category of adjunctions and the 2-category of monads. One is related to the Kleisli adjunction and the other to the Eilenberg-Moore adjunction for a given monad.Since any 2-adjunction induces certain natural isomorphisms of categories, these can be used to classify bijections and isomorphisms for certain structures in monad theory. In particular, one important example of a structure, lying in the 2-category of adjunctions, where this procedure can be applied to is that of a lifting. Therefore, a lifting can be characterized by the associated monad structure,lying in the 2-category of monads, through the respective 2-adjunction. The same can be said for Kleisli extensions.Several authors have been discovered this type of bijections and isomorphisms but these pair of 2-adjunctions can collect them all at once with an extra property, that of naturality.Mon, 31 Dec 2018 20:30:00 +0100The category of generalized crossed modules
http://cgasa.sbu.ac.ir/article_69897_10418.html
In the definition of a crossed module $(T,G,rho)$, the actions of the group $T$ and $G$ on themselves are given by conjugation. In this paper, we consider these actions to be arbitrary and thus generalize the concept of ordinary crossed module. Therefore, we get the category ${bf GCM}$, of all generalized crossed modules and generalized crossed module morphisms between them, and investigate some of its categorical properties. In particular, we study the relations between epimorphisms and the surjective morphisms, and thus generalize the corresponding results of the category of (ordinary) crossed modules. By generalizing the conjugation action, we can find out what is the superiority of the conjugation to other actions. Also, we can find out a generalized crossed module with which other actions (other than the conjugation) has the properties same as a crossed module.Mon, 31 Dec 2018 20:30:00 +0100Persian Abstracts, Vol. 10, No. 1.
http://cgasa.sbu.ac.ir/article_81555_10418.html
Mon, 31 Dec 2018 20:30:00 +0100(r,t)-injectivity in the category S-Act
http://cgasa.sbu.ac.ir/article_76601_0.html
In this paper, we show that injectivity with respect to the class $mathcal{D}$ of dense monomorphisms of an idempotent and weakly hereditary closure operator of an arbitrary category well-behaves. Indeed, if $mathcal{M}$ is a subclass of monomorphisms, $mathcal{M}cap mathcal{D}$-injectivity well-behaves. We also introduce the notion of $(r,t)$-injectivity in the category {bf S-Act}, where $r$ and $t$ are Hoehnke radicals, and discuss whether this kind of injectivity well-behaves.Wed, 31 Oct 2018 20:30:00 +0100Intersection graphs associated with semigroup acts
http://cgasa.sbu.ac.ir/article_76602_0.html
The intersection graph $mathbbm{Int}(A)$ of an $S$-act $A$ over a semigroup $S$ is an undirected simple graph whose vertices are non-trivial subacts of $A$, and two distinct vertices are adjacent if and only if they have a non-empty intersection. In this paper, we study some graph-theoretic properties of $mathbbm{Int}(A)$ in connection to some algebraic properties of $A$. It is proved that the finiteness of each of the clique number, the chromatic number, and the degree of some or all vertices in $mathbbm{Int}(A)$ is equivalent to the finiteness of the number of subacts of $A$. Finally, we determine the clique number of the graphs of certain classes of $S$-acts.Wed, 31 Oct 2018 20:30:00 +0100On Semi Weak Factorization Structures
http://cgasa.sbu.ac.ir/article_76603_0.html
In this article the notions of semi weak orthogonality and semi weak factorization structure in acategory $calx$ are introduced. Then the relationship between semi weak factorization structures and quasi right (left) and weak factorization structures is given. The main result is a characterization of semi weak orthogonality, factorization of morphisms, and semi weak factorization structures by natural isomorphisms.Wed, 31 Oct 2018 20:30:00 +0100Another proof of Banaschewski's surjection theorem
http://cgasa.sbu.ac.ir/article_76726_0.html
We present a new proof of Banaschewski's theorem stating that the completion lift of a uniform surjection is a surjection. The new procedure allows to extend the fact (and, similarly, the related theorem on closed uniform sublocales of complete uniform frames) to quasi-uniformities ("not necessarily symmetric uniformities"). Further, we show how a (regular) Cauchy point on a closed uniform sublocale can be extended to a (regular) Cauchy point on the larger (quasi-)uniform frame.Fri, 02 Nov 2018 20:30:00 +0100On exact category of $(m, n)$-ary hypermodules
http://cgasa.sbu.ac.ir/article_80792_0.html
We introduce and study category of $(m, n)$-ary hypermodules as a generalization of the category of $(m, n)$-modules as well as the category of classical modules. Also, we study various kinds of morphisms. Especially, we characterize monomorphisms and epimorphisms in this category. We will proceed to study the fundamental relation on $(m, n)$-hypermodules, as an important tool in the study of algebraic hyperstructures and prove that this relation is really functorial, that is, we introduce the fundamental functor from the category of $(m, n)$-hypermodules to the category $(m, n)$-modules and prove that it preserves monomorphisms. Finally, we prove that the category of $(m, n)$-hypermodules is an exact category, and, hence, it generalizes the classical case.Mon, 24 Dec 2018 20:30:00 +0100On $GPW$-Flat Acts
http://cgasa.sbu.ac.ir/article_82637_0.html
In this article, we present $GPW$-flatness property of acts over monoids, which is a generalization of principal weak flatness. We say that a right $S$-act $A_{S}$ is $GPW$-flat if for every $s in S$, there exists a natural number $n = n_ {(s, A_{S})} in mathbb{N}$ such that the functor $A_{S} otimes {}_{S}- $ preserves the embedding of the principal left ideal ${}_{S}(Ss^n)$ into ${}_{S}S$. We show that a right $S$-act $A_{S}$ is $GPW$-flat if and only if for every $s in S$ there exists a natural number $n = n_{(s, A_{S})} in mathbb{N}$ such that the corresponding $varphi$ is surjective for the pullback diagram $P(Ss^n, Ss^n, iota, iota, S)$, where $iota : {}_{S}(Ss^n) rightarrow {}_{S}S$ is a monomorphism of left $S$-acts. Also we give some general properties and a characterization of monoids for which this condition of their acts implies some other properties and vice versa.Sun, 10 Feb 2019 20:30:00 +0100Completeness results for metrized rings and lattices
http://cgasa.sbu.ac.ir/article_82638_0.html
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper {em radical} ideals (for example, ${0})$ that are closed under the natural metric, but has no {em prime} ideal closed under that metric; hence closed radical ideals are not, in general, intersections of closed prime ideals. Moreover, $B$ is known to be complete in its metric. Together, these facts answer a question posed by J.,Gleason. From this example, rings of arbitrary characteristic with the same properties are obtained. The result that $B$ is complete in its metric is generalized to show that if $L$ is a lattice given with a metric satisfying identically {em either} the inequality $d(xvee y,,xvee z)leq d(y,z)$ {em or} the inequality $d(xwedge y,linebreak[2],xwedge z)leq d(y,z),$ and if in $L$ every increasing Cauchy sequence converges and every decreasing Cauchy sequence converges, then every Cauchy sequence in $L$ converges; that is, $L$ is complete as a metric space. We show by example that if the above inequalities are replaced by the weaker conditions $d(x,,xvee y)leq d(x,y),$ respectively $d(x,linebreak[2],xwedge y)linebreak[2]leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.Sun, 10 Feb 2019 20:30:00 +0100A convex combinatorial property of compact sets in the plane and its roots in lattice theory
http://cgasa.sbu.ac.ir/article_82639_0.html
K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ for the more general case where $,mathcal U_0$ and $,mathcal U_1$ are compact sets in the plane such that $,mathcal U_1$ is obtained from $,mathcal U_0$ by a positive homothety or by a translation. Also, we give a short survey to show how lattice theoretical antecedents, including a series of papers on planar semimodular lattices by G. Gr"atzer and E. Knapp, lead to our result.Sun, 10 Feb 2019 20:30:00 +0100Classification of monoids by Condition $(PWP_{ssc})$
http://cgasa.sbu.ac.ir/article_85729_0.html
Condition $(PWP)$ which was introduced in (Laan, V., {it Pullbacks and flatness properties of acts I}, Commun. Algebra, 29(2) (2001), 829-850), is related to flatness concept of acts over monoids. Golchin and Mohammadzadeh in ({it On Condition $(PWP_E)$}, Southeast Asian Bull. Math., 33 (2009), 245-256) introduced Condition $(PWP_E)$, such that Condition $(PWP)$ implies it, that is, Condition $(PWP_E)$ is a generalization of Condition $(PWP)$. In this paper we introduce Condition $(PWP_{ssc})$, which is much easier to check than Conditions $(PWP)$ and $(PWP_E)$ and does not imply them. Also principally weakly flat is a generalization of this condition. At first, general properties of Condition $(PWP_{ssc})$ will be given. Finally a classification of monoids will be given for which all (cyclic, monocyclic) acts satisfy Condition $(PWP_{ssc})$ and also a classification of monoids $S$ will be given for which all right $S$-acts satisfying some other flatness properties have Condition $(PWP_{ssc})$.Fri, 05 Apr 2019 19:30:00 +0100Frankl's Conjecture for a subclass of semimodular lattices
http://cgasa.sbu.ac.ir/article_85730_0.html
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices having breadth at most two. We provide a very short proof of the Conjecture for the class of lattices having breadth at most two. This generalizes the results of Joshi, Waphare and Kavishwar as well as Cz'edli and Schmidt.Fri, 05 Apr 2019 19:30:00 +0100