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The structure of completely meet irreducible congruences in strongly Fregean algebras 强代数中完全满足不可约同余的结构
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-07-13 DOI: 10.1007/s00012-022-00787-0
K. Słomczyńska
{"title":"The structure of completely meet irreducible congruences in strongly Fregean algebras","authors":"K. Słomczyńska","doi":"10.1007/s00012-022-00787-0","DOIUrl":"https://doi.org/10.1007/s00012-022-00787-0","url":null,"abstract":"","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51690177","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
The structure of completely meet irreducible congruences in strongly Fregean algebras 强Fregean代数中完全满足不可约同余的结构
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-07-13 DOI: 10.1007/s00012-022-00787-0
Katarzyna Słomczyńska
{"title":"The structure of completely meet irreducible congruences in strongly Fregean algebras","authors":"Katarzyna Słomczyńska","doi":"10.1007/s00012-022-00787-0","DOIUrl":"10.1007/s00012-022-00787-0","url":null,"abstract":"<div><p>A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50477842","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: 1
Minimum proper extensions in some lattices of subalgebras 子代数格中的最小固有扩展
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-07-07 DOI: 10.1007/s00012-022-00784-3
Anthony W. Hager, Brian Wynne
{"title":"Minimum proper extensions in some lattices of subalgebras","authors":"Anthony W. Hager,&nbsp;Brian Wynne","doi":"10.1007/s00012-022-00784-3","DOIUrl":"10.1007/s00012-022-00784-3","url":null,"abstract":"<div><p>Let <span>({mathcal {A}})</span> be a class of algebras with <span>(I, A in {mathcal {A}})</span>. We interpret the lattice-theoretic “strictly meet irreducible/cover” situation <span>(B &lt; C)</span> in lattices of the form <span>(S_{{mathcal {A}}}(I,A))</span> of all subalgebras of <i>A</i> containing <i>I</i>, where we call such <span>(B &lt; C)</span> a <i>minimum proper extension</i> (mpe), and show that this means <i>B</i> is maximal in <span>(S_{{mathcal {A}}}(I,A))</span> for not containing some <span>(r in A)</span> and <i>C</i> is generated by <i>B</i> and <i>r</i>. For the class <span>({mathcal {G}})</span> of groups, we determine the mpe’s in <span>(S_{{mathcal {G}}}({0},{mathbb {Q}}))</span> using invariants of Beaumont and Zuckerman and show that these (plus utilization of a Hamel basis) determine the mpe’s in <span>(S_{{mathcal {G}}}({0},{mathbb {R}}))</span>. Finally, we show that the latter yield some (not all) of the minimum proper essential extensions in <span>(mathbf {W}^{*})</span>, the category of Archimedean <span>(ell )</span>-groups with strong order unit and unit-preserving <span>(ell )</span>-group homomorphisms.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43308755","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
Localic transitivity 局部及物性
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-28 DOI: 10.1007/s00012-022-00783-4
Ali Akbar Estaji, Maryam Robat Sarpoushi, Ali Barzanouni
{"title":"Localic transitivity","authors":"Ali Akbar Estaji,&nbsp;Maryam Robat Sarpoushi,&nbsp;Ali Barzanouni","doi":"10.1007/s00012-022-00783-4","DOIUrl":"10.1007/s00012-022-00783-4","url":null,"abstract":"<div><p>For a dynamical system (<i>X</i>, <i>f</i>), the notion of <i>topological transitivity</i> has been studied by some researchers. There are several definitions of this property, and it is part of the folklore of dynamical systems that under some hypotheses, they are equivalent. In this paper, our aim is to introduce and study some properties of topological transitivity in pointfree topology, for which we first need to define in a way what makes them conservative extensions of topological transitivity defined by G.D. Birkhoff. We describe the way the different properties are related to each other in pointfree topology.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46921771","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 finite axiomatization of positive MV-algebras 正mv -代数的有限公理化
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-28 DOI: 10.1007/s00012-022-00776-3
Marco Abbadini, Peter Jipsen, Tomáš Kroupa, Sara Vannucci
{"title":"A finite axiomatization of positive MV-algebras","authors":"Marco Abbadini,&nbsp;Peter Jipsen,&nbsp;Tomáš Kroupa,&nbsp;Sara Vannucci","doi":"10.1007/s00012-022-00776-3","DOIUrl":"10.1007/s00012-022-00776-3","url":null,"abstract":"<div><p>Positive MV-algebras are the subreducts of MV-algebras with respect to the signature <span>({oplus , odot , vee , wedge , 0, 1})</span>. We provide a finite quasi-equational axiomatization for the class of such algebras.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45643920","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: 2
On relatively elementary definability of graph classes in the class of semigroups 半群类中图类的相对初等可定义性
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-27 DOI: 10.1007/s00012-022-00780-7
Vladimir A. Molchanov
{"title":"On relatively elementary definability of graph classes in the class of semigroups","authors":"Vladimir A. Molchanov","doi":"10.1007/s00012-022-00780-7","DOIUrl":"10.1007/s00012-022-00780-7","url":null,"abstract":"<div><p>Based on the previously obtained concrete characterization of the endomorphism semigroups of quasi-acyclic reflexive graphs we prove the relatively elementary definability of the class of such graphs in the class of all semigroups. It will permit us to investigate for such graphs the abstract representation problem for the endomorphism semigroups of graphs and the problem of elementary definability of graphs by their endomorphism semigroups.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45602201","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 relatively finite-to-finite universal but not Q-universal quasivariety 一个相对有限到有限的泛但不是q -泛的拟变
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-27 DOI: 10.1007/s00012-022-00782-5
M. E. Adams, W. Dziobiak, H. P. Sankappanavar
{"title":"A relatively finite-to-finite universal but not Q-universal quasivariety","authors":"M. E. Adams,&nbsp;W. Dziobiak,&nbsp;H. P. Sankappanavar","doi":"10.1007/s00012-022-00782-5","DOIUrl":"10.1007/s00012-022-00782-5","url":null,"abstract":"<div><p>It was proved by the authors that the quasivariety of quasi-Stone algebras <span>(mathbf {Q}_{mathbf {1,2}})</span> is finite-to-finite universal relative to the quasivariety <span>(mathbf {Q}_{mathbf {2,1}})</span> contained in <span>(mathbf {Q}_{mathbf {1,2}})</span>. In this paper, we prove that <span>(mathbf {Q}_{mathbf {1,2}})</span> is not Q-universal. This provides a positive answer to the following long standing open question: Is there a quasivariety that is relatively finite-to-finite universal but is not Q-universal?</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-022-00782-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41263333","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
Difference–restriction algebras of partial functions: axiomatisations and representations 偏函数的差限制代数：公理化与表示
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-27 DOI: 10.1007/s00012-022-00775-4
Célia Borlido, Brett McLean
{"title":"Difference–restriction algebras of partial functions: axiomatisations and representations","authors":"Célia Borlido,&nbsp;Brett McLean","doi":"10.1007/s00012-022-00775-4","DOIUrl":"10.1007/s00012-022-00775-4","url":null,"abstract":"<div><p>We investigate the representation and complete representation classes for algebras of partial functions with the signature of relative complement and domain restriction. We provide and prove the correctness of a finite equational axiomatisation for the class of algebras representable by partial functions. As a corollary, the same equations axiomatise the algebras representable by injective partial functions. For complete representations, we show that a representation is meet complete if and only if it is join complete. Then we show that the class of completely representable algebras is precisely the class of atomic and representable algebras. As a corollary, the same properties axiomatise the class of algebras completely representable by injective partial functions. The universal-existential-universal axiomatisation this yields for these complete representation classes is the simplest possible, in the sense that no existential-universal-existential axiomatisation exists.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46003188","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: 4
Spectral properties of cBCK-algebras cBCK-代数的谱性质
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-27 DOI: 10.1007/s00012-022-00779-0
C. Matthew Evans
{"title":"Spectral properties of cBCK-algebras","authors":"C. Matthew Evans","doi":"10.1007/s00012-022-00779-0","DOIUrl":"10.1007/s00012-022-00779-0","url":null,"abstract":"<div><p>In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44514969","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: 2
The number fields that are ({O}^{*})-fields $$｛O｝^｛*｝$$字段的数字字段
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-06-27 DOI: 10.1007/s00012-022-00781-6
Jingjing Ma
{"title":"The number fields that are ({O}^{*})-fields","authors":"Jingjing Ma","doi":"10.1007/s00012-022-00781-6","DOIUrl":"10.1007/s00012-022-00781-6","url":null,"abstract":"<div><p>Using the theory on infinite primes of fields developed by Harrison in [2], the necessary and sufficient conditions are proved for real number fields to be <span>(O^{*})</span>-fields, and many examples of <span>(O^{*})</span>-fields are provided.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43830966","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: 3
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