Algebra Universalis最新文献

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"Submaximal clones over a three-element set up to minor-equivalence","authors":"Albert Vucaj,&nbsp;Dmitriy Zhuk","doi":"10.1007/s00012-024-00852-w","DOIUrl":"10.1007/s00012-024-00852-w","url":null,"abstract":"<div><p>We study clones modulo minor homomorphisms, which are mappings from one clone to another preserving arities of operations and respecting permutation and identification of variables. Minor-equivalent clones satisfy the same sets of identities of the form <span>(f(x_1,dots ,x_n)approx g(y_1,dots ,y_m))</span>, also known as minor identities, and therefore share many algebraic properties. Moreover, it was proved that the complexity of the <span>({text {CSP}})</span> of a finite structure <span>(mathbb {A})</span> only depends on the set of minor identities satisfied by the polymorphism clone of <span>(mathbb {A})</span>. In this article we consider the poset that arises by considering all clones over a three-element set with the following order: we write <span>(mathcal {C} {preceq _{textrm{m}}} mathcal {D})</span> if there exists a minor homomorphism from <span>(mathcal {C})</span> to <span>(mathcal {D})</span>. We show that the aforementioned poset has only three submaximal elements.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00852-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140155945","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}

{"title":"On k-ary parts of maximal clones","authors":"Dragan Mašulović,&nbsp;Maja Pech","doi":"10.1007/s00012-024-00851-x","DOIUrl":"10.1007/s00012-024-00851-x","url":null,"abstract":"<div><p>The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine <i>k</i>-ary parts of the clones and their mutual inclusions. In this paper we study <i>k</i>-ary parts of maximal clones, for <span>(kgeqslant 2,)</span> building on the already known results for their unary parts. It turns out that the poset of <i>k</i>-ary parts of maximal clones defined by central relations contains long chains.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140155940","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}

{"title":"Generalized quasiorders and the Galois connection ({textbf {End}})–(varvec{{{,textrm{gQuord},}}})","authors":"Danica Jakubíková-Studenovská,&nbsp;Reinhard Pöschel,&nbsp;Sándor Radeleczki","doi":"10.1007/s00012-024-00850-y","DOIUrl":"10.1007/s00012-024-00850-y","url":null,"abstract":"<div><p>Equivalence relations or, more general, quasiorders (i.e., reflexive and transitive binary relations) <span>(varrho )</span> have the property that an <i>n</i>-ary operation <i>f</i> preserves <span>(varrho ,)</span> i.e., <i>f</i> is a polymorphism of <span>(varrho ,)</span> if and only if each translation (i.e., unary polynomial function obtained from <i>f</i> by substituting constants) preserves <span>(varrho ,)</span> i.e., it is an endomorphism of <span>(varrho .)</span> We introduce a wider class of relations—called generalized quasiorders—of arbitrary arities with the same property. With these generalized quasiorders we can characterize all algebras whose clone of term operations is determined by its translations by the above property, what generalizes affine complete algebras. The results are based on the characterization of so-called u-closed monoids (i.e., the unary parts of clones with the above property) as Galois closures of the Galois connection <span>({textrm{End}})</span>–<span>({{,textrm{gQuord},}},)</span> i.e., as endomorphism monoids of generalized quasiorders. The minimal u-closed monoids are described explicitly.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00850-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140155933","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}

{"title":"Some further results on pointfree convex geometry","authors":"Changchun Xia","doi":"10.1007/s00012-024-00847-7","DOIUrl":"10.1007/s00012-024-00847-7","url":null,"abstract":"<div><p>Inspired by locale theory, pointfree convex geometry was first proposed and studied by Yoshihiro Maruyama. In this paper, we shall continue to his work and investigate the related topics on pointfree convex spaces. Concretely, the following results are obtained: (1) A Hofmann–Lawson-like duality for pointfree convex spaces is established. (2) The <span>(mathcal {M})</span>-injective objects in the category of <span>(S_0)</span>-convex spaces are proved precisely to be sober convex spaces, where <span>(mathcal {M})</span> is the class of strict maps of convex spaces; (3) A convex space <i>X</i> is sober iff there never exists a nontrivial identical embedding <span>(i:Xhookrightarrow Y)</span> such that its dualization is an isomorphism, and a convex space <i>X</i> is <span>(S_D)</span> iff there never exists a nontrivial identical embedding <span>(k:Yhookrightarrow X)</span> such that its dualization is an isomorphism. (4) A dual adjunction between the category <span>(textbf{CLat}_D)</span> of continuous lattices with continuous <i>D</i>-homomorphisms and the category <span>(textbf{CS}_D)</span> of <span>(S_D)</span>-convex spaces with <i>CP</i>-maps is constructed, which can further induce a dual equivalence between <span>(textbf{CS}_D)</span> and a subcategory of <span>(textbf{CLat}_D)</span>; (5) The relationship between the quotients of a continuous lattice <i>L</i> and the convex subspaces of <span>({textbf {cpt}}(L))</span> is investigated and the collection <span>({textbf {Alg}}({textbf {Q}}(L)))</span> of all algebraic quotients of <i>L</i> is proved to be an algebraic join-sub-complete lattice of <span>({textbf {Q}}(L))</span> of all quotients of <i>L</i>, where <span>({textbf {cpt}}(L))</span> denote the set of non-bottom compact elements of <i>L</i>. Furthermore, it is shown that <span>({textbf {Alg}}({textbf {Q}}(L)))</span> is isomorphic to the collection <span>({textbf {Sob}}(mathcal {P}({textbf {cpt}}(L))))</span> of all sober convex subspaces of <span>({textbf {cpt}}(L))</span>; (6) Several necessary and sufficient conditions for all convex subspaces of <span>({textbf {cpt}}(L))</span> to be sober are presented.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043960","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}

{"title":"Factor principal congruences and Boolean products in filtral varieties","authors":"Brian A. Davey,&nbsp;Miroslav Haviar","doi":"10.1007/s00012-024-00846-8","DOIUrl":"10.1007/s00012-024-00846-8","url":null,"abstract":"<div><p>Motivated by Haviar and Ploščica’s 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner’s Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00846-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140036271","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}

{"title":"Joins and meets in effect algebras","authors":"Grzegorz Bińczak,&nbsp;Joanna Kaleta,&nbsp;Andrzej Zembrzuski","doi":"10.1007/s00012-024-00844-w","DOIUrl":"10.1007/s00012-024-00844-w","url":null,"abstract":"<div><p>It is known that every effect algebra can be represented as the effect algebra of perspectivity classes of some <i>E</i>-test space. We describe when there exists join and meet of two perspectivity classes of events of some algebraic <i>E</i>-test space. Moreover we give the formula for join and meet of perspectivity classes mentioned above, using only tests. We obtain an example of finite, non-homogeneous effect algebra <i>E</i> such that sharp elements of <i>E</i> form a lattice, whereas <i>E</i> is not a lattice.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00844-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140017896","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}