### فهرست مطالب

• Volume:11 Issue:1, 2019
• Special Issue Dedicated to Prof. George A. Grätzer
• تاریخ انتشار: 1398/04/10
• تعداد عناوین: 11
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• برنارد بناشفسکی صفحات 19-32
• آزاده ایلاقی حسینی، سید شاهین موسوی، سید ناصر حسینی صفحات 33-56
• گابور شزدلی، آرپاد کوروسا صفحات 57-92
• عبدالقاسم کریمی فیض آبادی صفحات 93-112
• دارمنند بابولار، جورج پیکادو، و الس پولتر صفحات 113-130
• عبدالحسین دلفان، حمید رسولی، ابوالفضل تهرانیان صفحات 131-148
• جورج ام. برگمن صفحات 149-168
• مهدیه حدادی، سید مجتبی ناصر شیخ الاسلامی صفحات 169-196
• وینایاک جوشی و بی.ان. وافیر صفحات 197-206
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• Gábor Czédli Pages 1-10

It is a great honor for me to write a few introductory words to the present volume of CGASA \bf{dedicated to Professor George A. Grätzer}. The occasion for this dedication is that we are celebrating two anniversaries in 2018 related to him. Namely,

Keywords: George Grätzer, lattice theory, Grätzer--Schmidt Theorem
• Gábor Czédli Pages 11-18

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.

Keywords: George A. Grätzer, Interview
• Bernhard Banaschewski Pages 19-32

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.

Keywords: Completely regular frames, zero dimensional frames, completely regular σ-frames, zero dimensional σ-frames, real-valued continuous functions, integer-valued continuous functions on frames
• Azadeh Ilaghi, Hosseini, Seyed Shahin Mousavi Mirkalai *, Naser Hosseini Pages 33-56

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.

Keywords: Quasi right (left) factorization structure, (semi weak) orthogonality, (semi weak), factorization structure
• Gábor Czédli *, Árpád Kurusa Pages 57-92

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.

Keywords: Congruence lattice, planar semimodular lattice, convex hull, compact set, linebreak circle, combinatorial geometry, abstract convex geometry, anti-exchange property
• Abolghasem Karimi Feizabadi Pages 93-112

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.

Keywords: Free object, ell-ring, ell-module, frame, cozero map, semi-cozero map, the F-Zariski topology, mathcalC-reticulation, lattice-valued map
• Dharmanand Baboolal, Jorge Picado, Ales Pultr Pages 113-130

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.

Keywords: Frame (locale), sublocale, uniform frame, quasi-uniform frame, uniform embedding, complete uniform frame, completion, Cauchy map, Cauchy filter, Cauchy complete
• Abdolhossein Delfan, Hamid Rasouli *, Abolfazl Tehranian Pages 131-148

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.

Keywords: S-act, intersection graph, chromatic number, clique number, weakly perfect graph
• George M. Bergmanorcid Pages 149-168

The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical</em> ideals (for example, ${0})$ that are closed under the natural metric, but has no 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. 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</em> the inequality $d(xvee y,,xvee z)leq d(y,z)$ 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. 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.

Keywords: Complete topological ring without closed prime ideals, measurable sets modulo sets of measure zero, lattice complete under a metric
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.
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.