发布求助
文献互助 智能选刊 最新文献

Algebra Universalis最新文献

筛选
英文 中文
A theorem of Mumford and Ramanujam for universal algebras 泛代数的Mumford和Ramanujam定理
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-08-07 DOI: 10.1007/s00012-022-00790-5
A. Clay, R. Padmanabhan
{"title":"A theorem of Mumford and Ramanujam for universal algebras","authors":"A. Clay,&nbsp;R. Padmanabhan","doi":"10.1007/s00012-022-00790-5","DOIUrl":"10.1007/s00012-022-00790-5","url":null,"abstract":"<div><p>A well-known result in quasigroup theory says that an associative quasigroup is a group, i.e. in quasigroups, associativity forces the existence of an identity element. The converse is, of course, far from true, as there are many, many non-associative loops. However, a remarkable theorem due to David Mumford and C.P. Ramanujam says that any projective variety having a binary morphism admitting a two-sided identity must be a group. Motivated by this result, we define a universal algebra (<i>A</i>; <i>F</i>) to be an <i>MR-algebra</i> if whenever a binary term function <i>m</i>(<i>x</i>, <i>y</i>) in the algebra admits a two-sided identity, then the reduct (<i>A</i>; <i>m</i>(<i>x</i>, <i>y</i>)) must be associative. Here we give some non-trivial varieties of quasigroups, groups, rings, fields and lattices which are MR-algebras. For example, every MR-quasigroup must be isotopic to a group, MR-groups are exactly the nilpotent groups of class 2, while commutative rings and complemented lattices are MR-algebras if and only if they are Boolean.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45442279","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
Choice-free duality for orthocomplemented lattices by means of spectral spaces 利用谱空间实现正交补格的无选择对偶
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-08-07 DOI: 10.1007/s00012-022-00789-y
Joseph McDonald, Kentarô Yamamoto
{"title":"Choice-free duality for orthocomplemented lattices by means of spectral spaces","authors":"Joseph McDonald,&nbsp;Kentarô Yamamoto","doi":"10.1007/s00012-022-00789-y","DOIUrl":"10.1007/s00012-022-00789-y","url":null,"abstract":"<div><p>The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological space <i>X</i> is compact if every subbasic open cover of <i>X</i> admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call <i>upper Vietoris orthospaces</i> in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45412955","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: 5
Monounary algebras containing subalgebras with meet-irreducible congruence lattice 含有满足不可约同余格的子代数的一元代数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-08-05 DOI: 10.1007/s00012-022-00786-1
Lucia Janičková
{"title":"Monounary algebras containing subalgebras with meet-irreducible congruence lattice","authors":"Lucia Janičková","doi":"10.1007/s00012-022-00786-1","DOIUrl":"10.1007/s00012-022-00786-1","url":null,"abstract":"<div><p>The system of all congruence lattices of all algebras with fixed base set <i>A</i> forms a lattice with respect to inclusion, denoted by <span>(mathcal {E}_A)</span>. Let <i>A</i> be finite. The meet-irreducible elements of <span>(mathcal {E}_A)</span> are congruence lattices of monounary algebras. We assume that (<i>A</i>, <i>f</i>) has a connected subalgebra <i>B</i> such that <i>B</i> contains at least 3 cyclic elements and <img> is meet-irreducible in <span>({mathcal {E}}_B)</span> and we prove several sufficient conditions under which <span>({{,mathrm{Con},}}(A, f))</span> is meet-irreducible in <span>({mathcal {E}}_A)</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45440928","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
Idempotent identities in f-rings f-环中的幂等恒等式
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-08-05 DOI: 10.1007/s00012-022-00792-3
Rawaa Hajji
{"title":"Idempotent identities in f-rings","authors":"Rawaa Hajji","doi":"10.1007/s00012-022-00792-3","DOIUrl":"10.1007/s00012-022-00792-3","url":null,"abstract":"<div><p>Let <i>A</i> be an Archimedean <i>f</i>-ring with identity and assume that <i>A</i> is equipped with another multiplication <span>(*)</span> so that <i>A</i> is an <i>f</i>-ring with identity <i>u</i>. Obviously, if <span>(*)</span> coincides with the original multiplication of <i>A</i> then <i>u</i> is idempotent in <i>A</i> (i.e., <span>(u^{2}=u)</span>). Conrad proved that the converse also holds, meaning that, it suffices to have <span>(u^{2}=u)</span> to conclude that <span>(*)</span> equals the original multiplication on <i>A</i>. The main purpose of this paper is to extend this result as follows. Let <i>A</i> be a (not necessary unital) Archimedean <i>f</i>-ring and <i>B</i> be an <span>(ell )</span>-subgroup of the underlaying <span>(ell )</span>-group of <i>A</i>. We will prove that if <i>B</i> is an <i>f</i>-ring with identity <i>u</i>, then the equality <span>(u^{2}=u)</span> is a necessary and sufficient condition for <i>B</i> to be an <i>f</i>-subring of <i>A</i>. As a key step in the proof of this generalization, we will show that the set of all <i>f</i>-subrings of <i>A</i> with the same identity has a smallest element and a greatest element with respect to the inclusion ordering. Also, we shall apply our main result to obtain a well known characterization of <i>f</i>-ring homomorphisms in terms of idempotent elements.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45349048","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
Lattices of retracts of direct products of two finite chains and notes on retracts of lattices 两个有限链直积的回缩格及关于回缩格的注记
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-08-02 DOI: 10.1007/s00012-022-00788-z
Gábor Czédli
{"title":"Lattices of retracts of direct products of two finite chains and notes on retracts of lattices","authors":"Gábor Czédli","doi":"10.1007/s00012-022-00788-z","DOIUrl":"10.1007/s00012-022-00788-z","url":null,"abstract":"<div><p>Ordered by set inclusion, the <i>retracts</i> of a lattice <i>L</i> together with the empty set form a bounded poset <span>(Ret (L))</span>. By a <i>grid</i> we mean the direct product of two non-singleton finite chains. We prove that if <i>G</i> is a grid, then <span>(Ret (G))</span> is a lattice. We determine the number of elements of <span>(Ret (G))</span>. Some easy properties of retracts, <i>retractions</i>, and their kernels called <i>retraction congruences</i> of (mainly distributive) lattices are found. Also, we present several examples, including a 12-element modular lattice <i>M</i> such that <span>(Ret (M))</span> is not a lattice.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50438720","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
The number of languages with maximum state complexity 具有最大状态复杂度的语言的数量
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-07-30 DOI: 10.1007/s00012-022-00785-2
Bjørn Kjos-Hanssen, Lei Liu
{"title":"The number of languages with maximum state complexity","authors":"Bjørn Kjos-Hanssen,&nbsp;Lei Liu","doi":"10.1007/s00012-022-00785-2","DOIUrl":"10.1007/s00012-022-00785-2","url":null,"abstract":"<div><p>Câmpeanu and Ho (2004) determined the maximum finite state complexity of finite languages, building on work of Champarnaud and Pin (1989). They stated that it is very difficult to determine the number of maximum-complexity languages. Here we give a formula for this number. We also generalize their work from languages to functions on finite sets.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-022-00785-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46646430","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: 1
Improved bounds on the size of the smallest representation of relation algebra (32_{65}) 关系代数最小表示大小的改进界（32_｛65｝）
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-07-30 DOI: 10.1007/s00012-022-00791-4
Jeremy F. Alm, Michael Levet, Saeed Moazami, Jorge Montero-Vallejo, Linda Pham, Dave Sexton, Xiaonan Xu
{"title":"Improved bounds on the size of the smallest representation of relation algebra (32_{65})","authors":"Jeremy F. Alm,&nbsp;Michael Levet,&nbsp;Saeed Moazami,&nbsp;Jorge Montero-Vallejo,&nbsp;Linda Pham,&nbsp;Dave Sexton,&nbsp;Xiaonan Xu","doi":"10.1007/s00012-022-00791-4","DOIUrl":"10.1007/s00012-022-00791-4","url":null,"abstract":"<div><p>In this paper, we shed new light on the spectrum of the relation algebra we call <span>(A_{n})</span>, which is obtained by splitting the non-flexible diversity atom of <span>(6_{7})</span> into <i>n</i> symmetric atoms. Precisely, show that the minimum value in <span>(text {Spec}(A_{n}))</span> is at most <span>(2n^{6 + o(1)})</span>, which is the first polynomial bound and improves upon the previous bound due to Dodd and Hirsch (J Relat Methods Comput Sci 2:18–26, 2013). We also improve the lower bound to <span>(2n^{2} + 4n + 1)</span>, which is roughly double the trivial bound of <span>(n^{2} + 2n + 3)</span>. In the process, we obtain stronger results regarding <span>(text {Spec}(A_{2}) =text {Spec}(32_{65}))</span>. Namely, we show that 1024 is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50525850","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 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":"83 1","pages":""},"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":"83 3","pages":""},"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":"83 3","pages":""},"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
首页 上一页
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整，为了账户安全，请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
请完成安全验证×
相关产品
×
本文献相关产品
联系我们：info@booksci.cn Book学术提供免费学术资源搜索服务，方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号：481959085
Book学术官方微信