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Distributive lattices with strong endomorphism kernel property as direct sums 具有强自同态核性质的直接和分配格
Categories and General Algebraic Structures with Application Pub Date : 2020-10-01 DOI: 10.29252/cgasa.13.1.45
J. Guričan
{"title":"Distributive lattices with strong endomorphism kernel property as direct sums","authors":"J. Guričan","doi":"10.29252/cgasa.13.1.45","DOIUrl":"https://doi.org/10.29252/cgasa.13.1.45","url":null,"abstract":"Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem 2.8). We shall determine the structure of special elements (which are introduced after Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":" 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120829069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The function ring functors of pointfree topology revisited 重新讨论无点拓扑的函数环函子
Categories and General Algebraic Structures with Application Pub Date : 2019-07-01 DOI: 10.29252/CGASA.11.1.19
B. Banaschewski
{"title":"The function ring functors of pointfree topology revisited","authors":"B. Banaschewski","doi":"10.29252/CGASA.11.1.19","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.19","url":null,"abstract":"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.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123725724","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
(r,t)-injectivity in the category S-Act (r,t)- s类act的注入性
Categories and General Algebraic Structures with Application Pub Date : 2019-07-01 DOI: 10.29252/CGASA.11.1.169
M. Haddadi, Seyed Mojtaba Naser Sheykholislami
{"title":"(r,t)-injectivity in the category S-Act","authors":"M. Haddadi, Seyed Mojtaba Naser Sheykholislami","doi":"10.29252/CGASA.11.1.169","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.169","url":null,"abstract":"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.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125669770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Intersection graphs associated with semigroup acts 与半群行为相关的交图
Categories and General Algebraic Structures with Application Pub Date : 2019-07-01 DOI: 10.29252/CGASA.11.1.131
A. Delfan, H. Rasouli, A. Tehranian
{"title":"Intersection graphs associated with semigroup acts","authors":"A. Delfan, H. Rasouli, A. Tehranian","doi":"10.29252/CGASA.11.1.131","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.131","url":null,"abstract":"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.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132647356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On semi weak factorization structures 关于半弱分解结构
Categories and General Algebraic Structures with Application Pub Date : 2019-07-01 DOI: 10.29252/CGASA.11.1.33
A. Ilaghi-Hosseini, Seyed Shahin Mousavi Mirkalai, Naser Hosseini
{"title":"On semi weak factorization structures","authors":"A. Ilaghi-Hosseini, Seyed Shahin Mousavi Mirkalai, Naser Hosseini","doi":"10.29252/CGASA.11.1.33","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.33","url":null,"abstract":"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.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115781032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Frankl's Conjecture for a subclass of semimodular lattices 半模格子类的Frankl猜想
Categories and General Algebraic Structures with Application Pub Date : 2019-07-01 DOI: 10.29252/CGASA.11.1.197
Vinayak Joshi, B. Waphare
{"title":"Frankl's Conjecture for a subclass of semimodular lattices","authors":"Vinayak Joshi, B. Waphare","doi":"10.29252/CGASA.11.1.197","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.197","url":null,"abstract":"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 Czedli and Schmidt.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126484899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The categories of lattice-valued maps, Equalities, Free objects, and $mathcal C$-reticulation 格值映射、等式、自由对象和数学C -网的范畴
Categories and General Algebraic Structures with Application Pub Date : 2019-07-01 DOI: 10.29252/CGASA.11.1.93
A. K. Feizabadi
{"title":"The categories of lattice-valued maps, Equalities, Free objects, and $mathcal C$-reticulation","authors":"A. K. Feizabadi","doi":"10.29252/CGASA.11.1.93","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.93","url":null,"abstract":"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.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117009934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The notions of closedness and D-connectedness in quantale-valued approach spaces 量子值逼近空间中的闭性和d -连通性概念
Categories and General Algebraic Structures with Application Pub Date : 2018-11-02 DOI: 10.29252/CGASA.12.1.149
Muhammad Qasim, Samed Ozkan
{"title":"The notions of closedness and D-connectedness in quantale-valued approach spaces","authors":"Muhammad Qasim, Samed Ozkan","doi":"10.29252/CGASA.12.1.149","DOIUrl":"https://doi.org/10.29252/CGASA.12.1.149","url":null,"abstract":"In this paper, we characterize the local T0 and T1 separation axioms for quantale-valued gauge space, show how these concepts are related to each other and apply them to L-approach space and L-approach system. Furthermore, we give the characterization of a closed point and D-connectedness in quantale-valued gauge space. Finally, we compare all these concepts with other.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128164712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Completeness results for metrized rings and lattices 度量环和格的完备性结果
Categories and General Algebraic Structures with Application Pub Date : 2018-08-13 DOI: 10.29252/CGASA.11.1.149
G. Bergman
{"title":"Completeness results for metrized rings and lattices","authors":"G. Bergman","doi":"10.29252/CGASA.11.1.149","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.149","url":null,"abstract":"The Boolean ring $B$ of measurable subsets of the unit interval, modulo sets of measure zero, has proper radical ideals (e.g., ${0})$ that are closed under the natural metric, but has no prime ideals 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 corresponding properties are obtained. \u0000The 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; i.e., $L$ is complete as a metric space. \u0000We 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.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"35 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114101072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A convex combinatorial property of compact sets in the plane and its roots in lattice theory 平面上紧集的凸组合性质及其格理论中的根
Categories and General Algebraic Structures with Application Pub Date : 2018-07-10 DOI: 10.29252/CGASA.11.1.57
G'abor Cz'edli, 'Arp'ad Kurusa
{"title":"A convex combinatorial property of compact sets in the plane and its roots in lattice theory","authors":"G'abor Cz'edli, 'Arp'ad Kurusa","doi":"10.29252/CGASA.11.1.57","DOIUrl":"https://doi.org/10.29252/CGASA.11.1.57","url":null,"abstract":"K. Adaricheva and M. Bolat have recently proved that if $U_0$ and $U_1$ 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 $U_{1-k}$ is included in the convex hull of $U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles. Here we prove the existence of such a $j$ and $k$ for the more general case where $U_0$ and $U_1$ are compact sets in the plane such that $U_1$ is obtained from $U_0$ 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. Gratzer and E. Knapp, lead to our result.","PeriodicalId":170235,"journal":{"name":"Categories and General Algebraic Structures with Application","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122496699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 15
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