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Profinite bi-Heyting algebras 无限双heyting代数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-05-09 DOI: 10.1007/s00012-025-00892-w
Lydia Tasiou
{"title":"Profinite bi-Heyting algebras","authors":"Lydia Tasiou","doi":"10.1007/s00012-025-00892-w","DOIUrl":"10.1007/s00012-025-00892-w","url":null,"abstract":"<div><p>A poset <span>({mathbb {X}})</span> is said to be zigzag image-finite, if the least updownset (i.e., both an upset and a downset) containing <i>x</i> is finite, for all <span>(xin X.)</span> We show that a bi-Heyting algebra is profinite if and only if it is isomorphic to the lattice of upsets of a zigzag image-finite poset. Zigzag image-finite posets have the property of being disjoint unions of finite connected posets. Because of this, we equivalently show that a bi-Heyting algebra is profinite if and only if it is isomorphic to a direct product of simple finite bi-Heyting algebras.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-025-00892-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143930124","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
Compact ICA-topobooleans and the Smirnov compactification theorem 紧凑型 ICA 拓扑布尔和斯米尔诺夫紧凑化定理
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-04-07 DOI: 10.1007/s00012-025-00885-9
Ali Akbar Estaji, Rahimeh Pourkhandani, Mehdi Vatandoost
{"title":"Compact ICA-topobooleans and the Smirnov compactification theorem","authors":"Ali Akbar Estaji,&nbsp;Rahimeh Pourkhandani,&nbsp;Mehdi Vatandoost","doi":"10.1007/s00012-025-00885-9","DOIUrl":"10.1007/s00012-025-00885-9","url":null,"abstract":"<div><p>Recently, the concepts of topoframe and topoboolean have been introduced as a generalization of point-free topology, and the relation between topobooleans and complete I-contact algebras (ICAs) has been studied. In this paper, we first introduce the ICA-topoboolean <span>(B_{tau (C)})</span>, in which <span>(tau (C))</span> is induced from the complete ICA (<i>B</i>, <i>C</i>), and then characterize compact atomic ICA-topobooleans by their point clusters. As an example of the noncompact case, we determine all clusters of <span>(big ( mathcal {P}(mathbb {R}), Cbig ))</span>, an ICA on the Boolean algebra of the power set of real numbers <span>(mathbb {R})</span>. Finally, we generalize the Smirnov compactification theorem from proximity spaces to atomic ICA-topobooleans.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143793114","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
Nonelementary inclusive varieties of groups and semigroups 群和半群的非初等包容变种
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-04-04 DOI: 10.1007/s00012-025-00887-7
G. Mashevitzky
{"title":"Nonelementary inclusive varieties of groups and semigroups","authors":"G. Mashevitzky","doi":"10.1007/s00012-025-00887-7","DOIUrl":"10.1007/s00012-025-00887-7","url":null,"abstract":"<div><p>The class of identical inclusions was defined by E. S. Lyapin. This is the class of universal formulas which is situated strictly between identities and universal positive formulas. Classes of semigroups defined by identical inclusions are called inclusive varieties. Inclusive varieties that cannot be defined by the first order formulas are called nonelementary inclusive varieties. We study nonelementary inclusive varieties of groups, Clifford semigroups and nilsemigroups. In particular, a criterion for an inclusive variety to be nonelementary is found and limit nonelementary inclusive varieties of abelian groups are described. We also describe the upper semilattice of nonelementary inclusive varieties of finite abelian groups and prove that it is uncountable. We find an uncountable set of nonelementary inclusive varieties of nilpotent class 3 and nil class 2 finite commutative semigroups and a limit nonelementary inclusive variety of nilsemigroups. We consider completely regular semigroups in semigroup signature with an additional unary operation and nilsemigroups in semigroup signature with the additional constant 0.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-025-00887-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143778067","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 the structure of modal and tense operators on a boolean algebra 布尔代数上模态算子和时态算子的结构
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-04-04 DOI: 10.1007/s00012-025-00890-y
Guram Bezhanishvili, Andre Kornell
{"title":"On the structure of modal and tense operators on a boolean algebra","authors":"Guram Bezhanishvili,&nbsp;Andre Kornell","doi":"10.1007/s00012-025-00890-y","DOIUrl":"10.1007/s00012-025-00890-y","url":null,"abstract":"<div><p>We initiate the study of the poset <span>(mathcal{N}mathcal{O}(B))</span> of necessity operators on a boolean algebra <i>B</i>. We show that <span>(mathcal{N}mathcal{O}(B))</span> is a meet-semilattice that need not be distributive. However, when <i>B</i> is complete, <span>(mathcal{N}mathcal{O}(B))</span> is necessarily a frame, which is spatial iff <i>B</i> is atomic. In that case, <span>(mathcal{N}mathcal{O}(B))</span> is a locally Stone frame. Dual results hold for the poset <span>(mathcal{P}mathcal{O}(B))</span> of possibility operators. We also obtain similar results for the posets <span>(mathcal {TNO}(B))</span> and <span>(mathcal {TPO}(B))</span> of tense necessity and possibility operators on <i>B</i>. Our main tool is Jónsson-Tarski duality, by which such operators correspond to continuous and interior relations on the Stone space of <i>B</i>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143778068","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
Monadic ortholattices: completions and duality 一元正正交：补全和对偶性
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-04-03 DOI: 10.1007/s00012-025-00889-5
John Harding, Joseph McDonald, Miguel Peinado
{"title":"Monadic ortholattices: completions and duality","authors":"John Harding,&nbsp;Joseph McDonald,&nbsp;Miguel Peinado","doi":"10.1007/s00012-025-00889-5","DOIUrl":"10.1007/s00012-025-00889-5","url":null,"abstract":"<div><p>We show that the variety of monadic ortholattices is closed under MacNeille and canonical completions. In each case, the completion of <i>L</i> is obtained by forming an associated dual space <i>X</i> that is a monadic orthoframe. This is a set with an orthogonality relation and an additional binary relation satisfying certain conditions. For the MacNeille completion, <i>X</i> is formed from the non-zero elements of <i>L</i>, and for the canonical completion, <i>X</i> is formed from the proper filters of <i>L</i>. The corresponding completion of <i>L</i> is then obtained as the ortholattice of bi-orthogonally closed subsets of <i>X</i> with an additional operation defined through the binary relation of <i>X</i>. With the introduction of a suitable topology on an orthoframe, as was done by Goldblatt and Bimbó, we obtain a dual adjunction between the categories of monadic ortholattices and monadic orthospaces. A restriction of this dual adjunction provides a dual equivalence.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769776","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
Representable distributive quasi relation algebras 可表示分布拟关系代数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-04-03 DOI: 10.1007/s00012-025-00884-w
Andrew Craig, Claudette Robinson
{"title":"Representable distributive quasi relation algebras","authors":"Andrew Craig,&nbsp;Claudette Robinson","doi":"10.1007/s00012-025-00884-w","DOIUrl":"10.1007/s00012-025-00884-w","url":null,"abstract":"<div><p>We give a definition of representability for distributive quasi relation algebras (DqRAs). These algebras are a generalisation of relation algebras and were first described by Galatos and Jipsen (Algebra Univers 69:1–21, 2013). Our definition uses a construction that starts with a poset. The algebra is concretely constructed as the lattice of upsets of a partially ordered equivalence relation. The key to defining the three negation-like unary operations is to impose certain symmetry requirements on the partial order. Our definition of representable distributive quasi relation algebras is easily seen to be a generalisation of the definition of representable relations algebras by Jónsson and Tarski (AMS 54:89, 1948). We give examples of representable DqRAs and give a necessary condition for an algebra to be finitely representable. We leave open the questions of whether every DqRA is representable, and also whether the class of representable DqRAs forms a variety. Moreover, our definition provides many other opportunities for investigations in the spirit of those carried out for representable relation algebras.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-025-00884-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143769833","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
Right-preordered groups from a categorical perspective 从分类的角度看右序群
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-03-25 DOI: 10.1007/s00012-025-00886-8
Maria Manuel Clementino, Andrea Montoli
{"title":"Right-preordered groups from a categorical perspective","authors":"Maria Manuel Clementino,&nbsp;Andrea Montoli","doi":"10.1007/s00012-025-00886-8","DOIUrl":"10.1007/s00012-025-00886-8","url":null,"abstract":"<div><p>We study categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, studying some exactness properties, and showing that it is a quasivariety. We show that, from an algebraic point of view, the category of right-preordered groups shares several properties with the one of monoids. Moreover, we describe split extensions of right-preordered groups, showing in particular that semidirect products of ordered groups always have a natural right-preorder.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143698528","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
Reorienting quandle orbits 调整烛台轨道的方向
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-03-25 DOI: 10.1007/s00012-025-00883-x
Lorenzo Traldi
{"title":"Reorienting quandle orbits","authors":"Lorenzo Traldi","doi":"10.1007/s00012-025-00883-x","DOIUrl":"10.1007/s00012-025-00883-x","url":null,"abstract":"<div><p>Motivated by knot theory, it is natural to define the orienta-tion-reversal of a quandle orbit by inverting all the translations given by elements of that orbit. In this short note we observe that this natural notion is unsuited to medial quandles.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-025-00883-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143698529","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
Characterizations of U-frames and frames that are finitely a U-frame u型框架和有限u型框架的特征
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-03-25 DOI: 10.1007/s00012-025-00888-6
Batsile Tlharesakgosi
{"title":"Characterizations of U-frames and frames that are finitely a U-frame","authors":"Batsile Tlharesakgosi","doi":"10.1007/s00012-025-00888-6","DOIUrl":"10.1007/s00012-025-00888-6","url":null,"abstract":"<div><p>In this article, we give algebraic characterizations of <i>U</i>-frames in terms of ring-theoretic properties of the ring <span>(mathcal {R}L)</span> of real-valued continuous functions on a completely regular frame <i>L</i>. We show that a frame is a <i>U</i>-frame if and only if it is an <i>F</i>-frame and its Čech–Stone compactification is zero-dimensional. We will also introduce frames that are finitely a <i>U</i>-frame and we will characterize them in terms of ring-theoretic properties in <span>(mathcal {R}L)</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-025-00888-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143698527","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
Correction: Projectivity in (bounded) commutative integral residuated lattices 修正：（有界）交换积分残差格中的投影
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2025-01-04 DOI: 10.1007/s00012-024-00879-z
Paolo Aglianò, Sara Ugolini
{"title":"Correction: Projectivity in (bounded) commutative integral residuated lattices","authors":"Paolo Aglianò,&nbsp;Sara Ugolini","doi":"10.1007/s00012-024-00879-z","DOIUrl":"10.1007/s00012-024-00879-z","url":null,"abstract":"","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"86 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00879-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142925585","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
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