Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701The function ring functors of pointfree topology revisited193287117ENBernhard BanaschewskiDepartment of Mathematics and Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada.Journal Article20180910This 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701On semi weak factorization structures335676603ENAzadeh Ilaghi-HosseiniDepartment of Pure Mathematics, Faculty of Math and Computer, Shahid Bahonar University of KermanSeyed Shahin Mousavi MirkalaiDepartment of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran0000-0002-2904-7692Naser HosseiniDepartment of Pure Mathematics, Faculty of Math and Computers, Shahid Bahonar University of Kerman, Kerman, IranJournal Article20171221In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701A convex combinatorial property of compact sets in the plane and its roots in lattice theory579282639ENGábor CzédliBolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, H6720 Hungary0000-0001-9990-3573Árpád KurusaBolyai Institute, University of Szeged, Szeged, Aradi vértanúk tere 1, Hungary H6720Journal Article20180709K. 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.<br />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. <br />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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701The categories of lattice-valued maps, equalities, free objects, and $mathcal C$-reticulation9311287118ENAbolghasem Karimi FeizabadiDepartment of Mathematics, Gorgan Branch, Islamic Azad University, Gorgan, Iran.Journal Article20180921In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701Another proof of Banaschewski's surjection theorem11313076726ENDharmanand BaboolalSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4000, South Africa.Jorge PicadoDepartment of Mathematics
University of Coimbra
PORTUGALAles PultrDepartment of Applied Mathematics and ITI, MFF, Charles University,
Malostranske nam. 24, 11800 Praha 1, Czech RepublicJournal Article20180617We 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701Intersection graphs associated with semigroup acts13114876602ENAbdolhossein DelfanDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran,Hamid RasouliDepartment of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, IranAbolfazl TehranianDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranJournal Article20180504The 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701Completeness results for metrized rings and lattices14916882638ENGeorge M.BergmanUniversity of California, Berkeley0000-0003-4027-7293Journal Article20180810The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper <em>radical</em> ideals (for example, ${0})$ that are closed under the natural metric, but has no <em>prime</em> 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. <br />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</em> the inequality $d(xvee y,,xvee z)leq d(y,z)$ <em>or</em> 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. <br />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. <br />We end with two open questions.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701(r,t)-injectivity in the category $S$-Act16919676601ENMahdieh HaddadiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.Seyed Mojtaba Naser SheykholislamiDepartment of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.Journal Article20180305In 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.Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585311Special Issue Dedicated to Prof. George A. Grätzer20190701Frankl's Conjecture for a subclass of semimodular lattices19720685730ENVinayak JoshiDepartment of Mathematics, Savitribai Phule Pune University (Formerly, University of Pune) Ganeshkhind Road, Pune - 4110070000-0001-9105-4634Baloo WaphareDepartment of Mathematics, Savitribai Phule Pune University,
Pune-411007, India.Journal Article20180701 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.