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On the networks of large embeddings 关于大型嵌入网络
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-07-09 DOI: 10.1007/s00012-024-00862-8
Tuğba Aslan, Mohamed Khaled, Gergely Székely
{"title":"On the networks of large embeddings","authors":"Tuğba Aslan,&nbsp;Mohamed Khaled,&nbsp;Gergely Székely","doi":"10.1007/s00012-024-00862-8","DOIUrl":"10.1007/s00012-024-00862-8","url":null,"abstract":"<div><p>We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense, between two given algebras in the class in hand; with the possibility that this distance may take the value <span>(infty )</span>. We display a number of inspirational examples from different areas of algebra, e.g., group theory and monounary algebras, to show that this research direction can be quite remarkable.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00862-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Freudenthal and other compactifications of continuous frames 弗赖登塔尔法和其他连续框架压缩法
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-07-04 DOI: 10.1007/s00012-024-00857-5
Simo Mthethwa, Gugulethu Nogwebela
{"title":"The Freudenthal and other compactifications of continuous frames","authors":"Simo Mthethwa,&nbsp;Gugulethu Nogwebela","doi":"10.1007/s00012-024-00857-5","DOIUrl":"10.1007/s00012-024-00857-5","url":null,"abstract":"<div><p>The <i>N</i>-star compactifications of frames are the frame-theoretic counterpart of the <i>N</i>-point compactifications of locally compact Hausdorff spaces. A <span>(pi )</span>-compactification of a frame <i>L</i> is a compactification constructed using a special type of a basis called a <span>(pi )</span>-compact basis; the Freudenthal compactification is the largest <span>(pi )</span>-compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all <i>N</i>-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which <i>N</i>-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00857-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Unilinear residuated lattices: axiomatization, varieties and FEP 单线性残差网格：公理化、品种和 FEP
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-06-21 DOI: 10.1007/s00012-024-00856-6
Nikolaos Galatos, Xiao Zhuang
{"title":"Unilinear residuated lattices: axiomatization, varieties and FEP","authors":"Nikolaos Galatos,&nbsp;Xiao Zhuang","doi":"10.1007/s00012-024-00856-6","DOIUrl":"10.1007/s00012-024-00856-6","url":null,"abstract":"<div><p>We characterize all residuated lattices that have height equal to 3 and show that the variety they generate has continuum-many subvarieties. More generally, we study unilinear residuated lattices: their lattice is a union of disjoint incomparable chains, with bounds added. We we give two general constructions of unilinear residuated lattices, provide an axiomatization and a proof-theoretic calculus for the variety they generate, and prove the finite model property for various subvarieties.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A structure theorem for truncations on an Archimedean vector lattice 阿基米德向量网格截断的结构定理
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-06-21 DOI: 10.1007/s00012-024-00858-4
Karim Boulabiar
{"title":"A structure theorem for truncations on an Archimedean vector lattice","authors":"Karim Boulabiar","doi":"10.1007/s00012-024-00858-4","DOIUrl":"10.1007/s00012-024-00858-4","url":null,"abstract":"<div><p>Let <i>X</i> be an Archimedean vector lattice and <span>(X_{+})</span> denote the positive cone of <i>X</i>. A unary operation <span>(varpi )</span> on <span>(X_{+})</span> is called a truncation on <i>X</i> if </p><div><div><span>$$begin{aligned} xwedge varpi left( yright) =varpi left( xright) wedge yquad text {for all }x,yin X_{+}. end{aligned}$$</span></div></div><p>Let <span>(X^{u})</span> denote the universal completion of <i>X</i> with a distinguished weak element <span>(e&gt;0.)</span> It is shown that a unary operation <span>(varpi )</span> on <span>(X_{+})</span> is a truncation on <i>X</i> if and only if there exists an element <span>(uin X^{u})</span> and a component <i>p</i> of <i>e</i> such that </p><div><div><span>$$begin{aligned} pwedge u=0quad text {and}quad varpi left( xright) =px+uwedge x text {for all }xin X_{+}. end{aligned}$$</span></div></div><p>Here, <i>px</i> is the product of <i>p</i> and <i>x</i> with respect to the unique lattice-ordered multiplication in <span>(X^{u})</span> having <i>e</i> as identity. As an example of illustration, if <span>(varpi )</span> is a truncation on some <span>(L_{p}left( {mu } right) )</span>-space then there exists a measurable set <i>A</i> and a function <span>(uin L_{0}left( {mu } right) )</span> vanishing on <i>A</i> such that <span>(varpi left( xright) =1_{A}x+uwedge x)</span> for all <span>(xin L_{p}left( {mu } right) .)</span></p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Characterization of self-majorizing elements in Archimedean vector lattices 阿基米德向量网格中自约化元素的特征
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-06-19 DOI: 10.1007/s00012-024-00860-w
Zied Jbeli, Mohamed Ali Toumi
{"title":"Characterization of self-majorizing elements in Archimedean vector lattices","authors":"Zied Jbeli,&nbsp;Mohamed Ali Toumi","doi":"10.1007/s00012-024-00860-w","DOIUrl":"10.1007/s00012-024-00860-w","url":null,"abstract":"<div><p>In this paper, new purely topological approaches are furnished in order to characterize self-majorizing elements in an Archimedean vector lattice <i>A</i>. More precisely, it is shown that an element <span>(0&lt;fin A)</span> is a self-majorizing element if and only if every <i>f</i>-maximal order ideal of <i>A</i> is relatively uniformly closed. In addition, it is proved that self-majorizing elements are characterized via the hull–kernel topology on both the set of all proper prime order ideals <span>({mathcal {P}})</span> and on the set of all <i>g</i>-maximal order ideals <span>({mathcal {Q}})</span> of <i>A</i>,  for all <span>(gin A^{+}.)</span> In fact, the set of all prime order ideals of <i>A</i> not containing <i>f</i> (respectively, the set of all <i>g</i>-maximal order ideals of <i>A</i> not containing <i>f</i>,  for all <span>(gin A^{+}))</span> is a closed with respect to the hull–kernel topology on <span>({mathcal {P}})</span> (respectively, on <span>({mathcal {Q}}))</span> if and only if <i>f</i> is a self-majorizing element in <i>A</i>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Slim patch lattices as absolute retracts and maximal lattices 作为绝对缩回和最大网格的薄片网格
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-06-17 DOI: 10.1007/s00012-024-00861-9
Gábor Czédli
{"title":"Slim patch lattices as absolute retracts and maximal lattices","authors":"Gábor Czédli","doi":"10.1007/s00012-024-00861-9","DOIUrl":"10.1007/s00012-024-00861-9","url":null,"abstract":"<div><p>We prove that <i>slim patch lattices</i> are exactly the <i>absolute retracts</i> with more than two elements for the category of slim semimodular lattices with length-preserving lattice embeddings as morphisms. Also, slim patch lattices are the same as the <i>maximal objects</i> <i>L</i> in this category such that <span>(|L|&gt;2.)</span> Furthermore, slim patch lattices are characterized as the <i>algebraically closed lattices</i> <i>L</i> in this category such that <span>(|L|&gt;2.)</span> Finally, we prove that if we consider <span>({0,1})</span>-preserving lattice homomorphisms rather than length-preserving ones, then the absolute retracts for the class of slim semimodular lattices are the at most 4-element boolean lattices.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Permutation invariant boolean states 排列不变布尔态
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-06-06 DOI: 10.1007/s00012-024-00859-3
Daniele Mundici
{"title":"Permutation invariant boolean states","authors":"Daniele Mundici","doi":"10.1007/s00012-024-00859-3","DOIUrl":"10.1007/s00012-024-00859-3","url":null,"abstract":"<div><p>We give a self-contained proof of the following result: Finitely additive probability measures (also known as “states”) of the free boolean algebra <span>({mathsf F}_omega )</span> over the free generating set <span>({X_1,X_2,ldots })</span> having the invariance property under finite permutations of the <span>(X_i)</span>, coincide with states lying in the closure of the set of convex combinations of product states of <span>({mathsf F}_omega )</span> in the vector space <span>(mathbb R^{{mathsf F}_omega })</span> equipped with the product topology. De Finetti’s celebrated exchangeability theorem can be easily recovered from our proof.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141378878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Relative lengths of Maltsev conditions 马尔采夫条件的相对长度
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-05-24 DOI: 10.1007/s00012-024-00855-7
Paolo Lipparini
{"title":"Relative lengths of Maltsev conditions","authors":"Paolo Lipparini","doi":"10.1007/s00012-024-00855-7","DOIUrl":"10.1007/s00012-024-00855-7","url":null,"abstract":"<div><p>We solve some problems about relative lengths of Maltsev conditions, in particular, we provide an affirmative answer to a classical problem raised by A. Day more than 50 years ago. In detail, both congruence distributive and congruence modular varieties admit Maltsev characterizations by means of the existence of a finite but variable number of appropriate terms. A. Day showed that from Jónsson terms <span>(t_0,ldots , t_n)</span> witnessing congruence distributivity it is possible to construct terms <span>(u_0,ldots , u _{2n-1} )</span> witnessing congruence modularity. We show that Day’s result about the number of such terms is sharp when <i>n</i> is even. We also deal with other kinds of terms, such as alvin, Gumm, directed, as well as with possible variations we will call “specular” and “defective”. All the results hold when restricted to locally finite varieties.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00855-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141102550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Measuring associativity: graph algebras of undirected graphs 测量关联性：无向图的图代数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-05-04 DOI: 10.1007/s00012-024-00854-8
Kamilla Kátai-Urbán, Tamás Waldhauser
{"title":"Measuring associativity: graph algebras of undirected graphs","authors":"Kamilla Kátai-Urbán,&nbsp;Tamás Waldhauser","doi":"10.1007/s00012-024-00854-8","DOIUrl":"10.1007/s00012-024-00854-8","url":null,"abstract":"<div><p>We study two measures of associativity for graph algebras of finite undirected graphs: the index of nonassociativity and (a variant of) the semigroup distance. We determine “almost associative” and “antiassociative” graphs with respect to both measures. It turns out that the antiassociative graphs are exactly the balanced complete bipartite graphs, no matter which of the two measures we consider. In the class of connected graphs the two notions of almost associativity are also equivalent.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00854-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140889388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On minimal semiring generating sets of finitely generated commutative parasemifields 论有限生成交换准域的最小配线生成集
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-04-08 DOI: 10.1007/s00012-024-00853-9
Vítězslav Kala, Lucien Šíma
{"title":"On minimal semiring generating sets of finitely generated commutative parasemifields","authors":"Vítězslav Kala,&nbsp;Lucien Šíma","doi":"10.1007/s00012-024-00853-9","DOIUrl":"10.1007/s00012-024-00853-9","url":null,"abstract":"<div><p>We study ideal-simple commutative semirings and summarize the results giving their classification, in particular when they are finitely generated. In the principal case of (para)semifields, we then consider their minimal number of generators and show that it grows linearly with the depth of an associated rooted forest.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140588297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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