Categories and General Algebraic Structures with ApplicationsCategories and General Algebraic Structures with Applications
http://cgasa.sbu.ac.ir/
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http://cgasa.sbu.ac.ir/
Feed provided by Categories and General Algebraic Structures with Applications. Click to visit.Cover for Vol. 11, No. 1.
http://cgasa.sbu.ac.ir/article_87123_11594.html
Sun, 30 Jun 2019 19:30:00 +0100Celebrating Professor George A. Grätzer
http://cgasa.sbu.ac.ir/article_87121_11594.html
It is a great honor for me to write a few introductory words to the present volume of CGASA dedicated to Professor George A. Grätzer. The occasion for this dedication is that we are celebrating two anniversaries in 2018 related to him. Namely,
(A1) it was 55 years ago that the Grätzer--Schmidt Theorem was published, and
(A2) it was 40 years ago that G. Grätzer's General Lattice Theory, which immediately became the Book in lattice theory for decades, appeared.Sun, 30 Jun 2019 19:30:00 +0100An interview with George A. Grätzer
http://cgasa.sbu.ac.ir/article_87120_11594.html
This interview was conducted in the second half of May, 2018. Both George Grätzer and the author were at home, in Toronto and Szeged, respectively. They communicated via a lot of e-mails and a few phone calls.Sun, 30 Jun 2019 19:30:00 +0100The function ring functors of pointfree topology revisited
http://cgasa.sbu.ac.ir/article_87117_11594.html
This paper establishes two new connections between the familiar function ring functor ${mathfrak R}$ on the category ${bf CRFrm}$ of completely regular frames and the category {bf CR}${mathbf sigma}${bf Frm} of completely regular $sigma$-frames as well as their counterparts for the analogous functor ${mathfrak Z}$ on the category {bf ODFrm} of 0-dimensional frames, given by the integer-valued functions, and for the related functors ${mathfrak R}^*$ and ${mathfrak Z}^*$ corresponding to the bounded functions. Further it is shown that some familiar facts concerning these functors are simple consequences of the present results.Sun, 30 Jun 2019 19:30:00 +0100On semi weak factorization structures
http://cgasa.sbu.ac.ir/article_76603_11594.html
In this article the notions of semi weak orthogonality and semi weak factorization structure in a category $mathcal X$ 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.Sun, 30 Jun 2019 19:30:00 +0100A convex combinatorial property of compact sets in the plane and its roots in lattice theory
http://cgasa.sbu.ac.ir/article_82639_11594.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. Grätzer and E. Knapp, lead to our result.Sun, 30 Jun 2019 19:30:00 +0100The categories of lattice-valued maps, equalities, free objects, and $\mathcal C$-reticulation
http://cgasa.sbu.ac.ir/article_87118_11594.html
In this paper, we study the concept of $mathcal C$-reticulation for the category $mathcal C$ whose objects are lattice-valued maps. The relation between the free objects in $mathcal C$ and the $mathcal C$-reticulation of rings and modules is discussed. Also, a method to construct $mathcal C$-reticulation is presented, in the case where $mathcal C$ is equational. Some relations between the concepts reticulation and satisfying equalities and inequalities are studied.Sun, 30 Jun 2019 19:30:00 +0100Another proof of Banaschewski's surjection theorem
http://cgasa.sbu.ac.ir/article_76726_11594.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.Sun, 30 Jun 2019 19:30:00 +0100Intersection graphs associated with semigroup acts
http://cgasa.sbu.ac.ir/article_76602_11594.html
The intersection graph $mathbb{Int}(A)$ 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 $mathbb{Int}(A)$ 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 $mathbb{Int}(A)$ 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.Sun, 30 Jun 2019 19:30:00 +0100Completeness results for metrized rings and lattices
http://cgasa.sbu.ac.ir/article_82638_11594.html
The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (for example, ${0})$ that are closed under the natural metric, but has no 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 either the inequality $d(xvee y,,xvee z)leq d(y,z)$ or the inequality $d(xwedge y,xwedge z)leq d(y,z),$ 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 $d(x,,xwedge y)leq d(x,y),$ the completeness conclusion can fail. We end with two open questions.Sun, 30 Jun 2019 19:30:00 +0100(r,t)-injectivity in the category $S$-Act
http://cgasa.sbu.ac.ir/article_76601_11594.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.Sun, 30 Jun 2019 19:30:00 +0100Frankl's Conjecture for a subclass of semimodular lattices
http://cgasa.sbu.ac.ir/article_85730_11594.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 Czédli and Schmidt.Sun, 30 Jun 2019 19:30:00 +0100Persian Abstracts, Vol. 11, No. 1.
http://cgasa.sbu.ac.ir/article_87122_11594.html
Sun, 30 Jun 2019 19:30:00 +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 +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 +0100From torsion theories to closure operators and factorization systems
http://cgasa.sbu.ac.ir/article_87116_0.html
Torsion theories are here extended to categories equipped with an ideal of 'null morphisms', or equivalently a full subcategory of 'null objects'. Instances of this extension include closure operators viewed as generalised torsion theories in a 'category of pairs', and factorization systems viewed as torsion theories in a category of morphisms. The first point has essentially been treated in [15].Thu, 04 Jul 2019 19:30:00 +0100Some aspects of cosheaves on diffeological spaces
http://cgasa.sbu.ac.ir/article_87119_0.html
We define a notion of cosheaves on diffeological spaces by cosheaves on the site of plots. This provides a framework to describe diffeological objects such as internal tangent bundles, the Poincar'{e} groupoids, and furthermore, homology theories such as cubic homology in diffeology by the language of cosheaves. We show that every cosheaf on a diffeological space induces a cosheaf in terms of the D-topological structure. We also study quasi-cosheaves, defined by pre-cosheaves which respect the colimit over covering generating families, and prove that cosheaves are quasi-cosheaves. Finally, a so-called quasi-v{C}ech homology with values in pre-cosheaves is established for diffeological spaces.Thu, 04 Jul 2019 19:30:00 +0100