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 +0100The Notion of Closedness and D-connectedness in Quantale-valued Approach Spaces
http://cgasa.sbu.ac.ir/article_87411_0.html
In this paper, we characterize local $T_{0}$ and $T_{1}$ quantale-valued gauge spaces, show how these concepts are related to each other and apply them to $mathcal{L}$-approach distance spaces and $mathcal{L}$-approach system spaces. Furthermore, we give the characterization of a closed point and $D$-connectedness in quantale-valued gauge spaces. Finally, we compare all these concepts to each other.Wed, 27 Nov 2019 20:30:00 +0100Witt rings of quadratically presentable fields
http://cgasa.sbu.ac.ir/article_87412_0.html
This paper introduces an approach to the axiomatic theory of quadratic forms based on {tmem{presentable}} partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of {tmem{quadratically presentable fields}}, that is, fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. In particular, Witt rings of symmetric bilinear forms over fields of arbitrary characteristics are isomorphic to Witt rings of suitably built quadratically presentable fields.Wed, 27 Nov 2019 20:30:00 +0100Separated finitely supported $Cb$-sets
http://cgasa.sbu.ac.ir/article_87413_0.html
The monoid $Cb$ of name substitutions and the notion of finitely supported $Cb$text{-}sets introduced by Pitts as a generalization of nominal sets. A simple finitely supported $Cb$-set is a one point extension of a cyclic nominal set. The support map of a simple finitely supported $Cb$-set is an injective map. Also, for every two distinct elements of a simple finitely supported $Cb$-set, there exists an element of the monoid $Cb$ which separates them by making just one of them into an element with the empty support.In this paper, we generalize these properties of simple finitely supported $Cb$-sets by modifying slightly the notion of the support map; defining the notion of $mathsf{2}$-equivariant support map; and introducing the notions of s-separated and z-separated finitely supported $Cb$-sets. We show that the notions of s-separated and z-separated coincide for a finitely supported $Cb$-set whose support map is $mathsf{2}$-equivariant. Among other results, we find a characterization of simple s-separated (or z-separated) finitely supported $Cb$-sets. Finally, we show that some subcategories of finitely supported $Cb$-sets with injective equivariant maps which constructed applying the defined notions are reflective.Wed, 27 Nov 2019 20:30:00 +0100Product preservation and stable units for reflections into idempotent subvarieties
http://cgasa.sbu.ac.ir/article_87414_0.html
We give a necessary and sufficient condition for the preservation of finite products by a reflection of a variety of universal algebras into an idempotent subvariety. It is also shown that simple and semi-left-exact reflections into subvarieties of universal algebras are the same. It then follows that a reflection of a variety of universal algebras into an idempotent subvariety has stable units if and only if it is simple and the above-mentioned condition holds.Wed, 27 Nov 2019 20:30:00 +0100$(m,n)$-Hyperideals in Ordered Semihypergroups
http://cgasa.sbu.ac.ir/article_87415_0.html
In this paper, first we introduce the notions of an $(m,n)$-hyperideal and a generalized $(m,n)$-hyperideal in an ordered semihypergroup, and then, some properties of these hyperideals are studied. Thereafter, we characterize $(m,n)$-regularity, $(m,0)$-regularity, and $(0,n)$-regularity of an ordered semihypergroup in terms of its $(m,n)$-hyperideals, $(m,0)$-hyperideals and $(0,n)$-hyperideals, respectively. The relations ${_mmathcal{I}}, mathcal{I}_n, mathcal{H}_m^n$, and $mathcal{B}_m^n$ on an ordered semihypergroup are, then, introduced. We prove that $mathcal{B}_m^n subseteq mathcal{H}_m^n$ on an ordered semihypergroup and provide a condition under which equality holds in the above inclusion. We also show that the $(m,0)$-regularity [$(0,n)$-regularity] of an element induce the $(m,0)$-regularity [$(0,n)$-regularity] of the whole $mathcal{H}_m^n$-class containing that element as well as the fact that $(m,n)$-regularity and $(m,n)$-right weakly regularity of an element induce the $(m,n)$-regularity and $(m,n)$-right weakly regularity of the whole $mathcal{B}_m^n$-class and $mathcal{H}_m^n$-class containing that element, respectively.Wed, 27 Nov 2019 20:30:00 +0100