{"title":"Natural dualities for varieties generated by finite positive MV-chains","authors":"Wolfgang Poiger","doi":"10.1007/s00012-024-00868-2","DOIUrl":"10.1007/s00012-024-00868-2","url":null,"abstract":"<div><p>We provide a simple natural duality for the varieties generated by the negation- and implication-free reduct of a finite MV-chain. We study these varieties through the dual equivalences thus obtained. For example, we fully characterize their algebraically closed, existentially closed and injective members. We also explore the relationship between this natural duality and Priestley duality in terms of distributive skeletons and Priestley powers.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202458","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":"Quasivarieties of algebras whose compact relative congruences are principal","authors":"Anvar M. Nurakunov","doi":"10.1007/s00012-024-00866-4","DOIUrl":"10.1007/s00012-024-00866-4","url":null,"abstract":"<div><p>A quasivariety <span>(mathfrak N)</span> is called <i>relative congruence principal</i> if, for every algebra <span>(Ain mathfrak N)</span>, every compact <span>(mathfrak N)</span>-congruence on <i>A</i> is a principal <span>(mathfrak N)</span>-congruence. We characterize relative congruence principal quasivarieties in terms of one identity and two quasi-identities. We will use the characterization to show that there exists a continuum of relative congruence principal quasivarieties of algebras of a signature <span>(sigma )</span>, provided <span>(sigma )</span> contains at least one operation of arity greater than 1. Several examples are provided.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202460","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":"Override and restricted union for partial functions","authors":"Tim Stokes","doi":"10.1007/s00012-024-00864-6","DOIUrl":"10.1007/s00012-024-00864-6","url":null,"abstract":"<div><p>The <i>override</i> operation <span>(sqcup )</span> is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions <i>f</i> and <i>g</i>, <span>(fsqcup g)</span> is the function with domain <span>({{,textrm{dom},}}(f)cup {{,textrm{dom},}}(g))</span> that agrees with <i>f</i> on <span>({{,textrm{dom},}}(f))</span> and with <i>g</i> on <span>({{,textrm{dom},}}(g) backslash {{,textrm{dom},}}(f))</span>. Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature <span>((sqcup ))</span>. But adding operations (such as <i>update</i>) to this minimal signature can lead to finite axiomatisations. For the functional signature <span>((sqcup ,backslash ))</span> where <span>(backslash )</span> is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define <span>(fcurlyvee g=(fsqcup g)cap (gsqcup f))</span> for all functions <i>f</i> and <i>g</i>; this is the largest domain restriction of the binary relation <span>(fcup g)</span> that gives a partial function. Now <span>(fcap g=fbackslash (fbackslash g))</span> and <span>(fsqcup g=fcurlyvee (fcurlyvee g))</span> for all functions <i>f</i>, <i>g</i>, so the signatures <span>((curlyvee ))</span> and <span>((sqcup ,cap ))</span> are both intermediate between <span>((sqcup ))</span> and <span>((sqcup ,backslash ))</span> in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00864-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202461","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":"(varvec{S})-preclones and the Galois connection (varvec{{}^{S}{}textrm{Pol}})–(varvec{{}^{S}{}textrm{Inv}}), Part I","authors":"Peter Jipsen, Erkko Lehtonen, Reinhard Pöschel","doi":"10.1007/s00012-024-00863-7","DOIUrl":"10.1007/s00012-024-00863-7","url":null,"abstract":"<div><p>We consider <i>S</i>-<i>operations</i> <span>(f :A^{n} rightarrow A)</span> in which each argument is assigned a <i>signum</i> <span>(s in S)</span> representing a “property” such as being order-preserving or order-reversing with respect to a fixed partial order on <i>A</i>. The set <i>S</i> of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of <i>S</i>-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all <i>S</i>-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of <i>S</i>-<i>preclone</i>. We introduce <i>S</i>-<i>relations</i> <span>(varrho = (varrho _{s})_{s in S})</span>, <i>S</i>-<i>relational clones</i>, and a preservation property (<img>), and we consider the induced Galois connection <span>({}^{S}{}textrm{Pol})</span>–<span>({}^{S}{}textrm{Inv})</span>. The <i>S</i>-preclones and <i>S</i>-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all <i>S</i>-preclones on <i>A</i>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00863-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568120","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 the networks of large embeddings","authors":"Tuğba Aslan, Mohamed Khaled, 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":null,"pages":null},"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}
{"title":"The Freudenthal and other compactifications of continuous frames","authors":"Simo Mthethwa, 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":null,"pages":null},"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}
{"title":"Unilinear residuated lattices: axiomatization, varieties and FEP","authors":"Nikolaos Galatos, 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":null,"pages":null},"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}
{"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>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":null,"pages":null},"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}
{"title":"Characterization of self-majorizing elements in Archimedean vector lattices","authors":"Zied Jbeli, 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<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":null,"pages":null},"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}
{"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|>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|>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":null,"pages":null},"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}