{"title":"Admissible subsets and completions of ordered algebras","authors":"Valdis Laan, Jianjun Feng, Xia Zhang","doi":"10.1007/s00012-023-00813-9","DOIUrl":"10.1007/s00012-023-00813-9","url":null,"abstract":"<div><p>We consider ordered universal algebras and give a construction of a join-completion for them using so-called <span>(mathscr {D})</span>-ideals. We show that this construction has a universal property that induces a reflector from a certain category of ordered algebras to the category of sup-algebras. Our results generalize several earlier known results about different ordered structures.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46428447","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":"Stone space partitions indexed by a poset","authors":"Andrew B. Apps","doi":"10.1007/s00012-023-00816-6","DOIUrl":"10.1007/s00012-023-00816-6","url":null,"abstract":"<div><p>Stone space partitions <span>({X_{p}mid pin P})</span> satisfying conditions like <span>(overline{X_{p}}=bigcup _{qleqslant p}X_{q})</span> for all <span>(pin P)</span>, where <i>P</i> is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of <span>(omega )</span>-categorical structures. A key concept for studying such partitions is that of a <i>p</i>-trim open set which meets precisely those <span>(X_{q})</span> for which <span>(qgeqslant p)</span>; for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition <span>({X_{r}mid rin [0,1]})</span> of the Cantor set such that <span>(overline{X_{r}}=bigcup _{sleqslant r}X_{s}text { for all }rin [0,1])</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49544457","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":"Topology of closure systems in algebraic lattices","authors":"Niels Schwartz","doi":"10.1007/s00012-023-00815-7","DOIUrl":"10.1007/s00012-023-00815-7","url":null,"abstract":"<div><p>Algebraic lattices are spectral spaces for the coarse lower topology. Closure systems in algebraic lattices are studied as subspaces. Connections between order theoretic properties of a closure system and topological properties of the subspace are explored. A closure system is algebraic if and only if it is a patch closed subset of the ambient algebraic lattice. Every subset <i>X</i> in an algebraic lattice <i>P</i> generates a closure system <span>(langle X rangle _P)</span>. The closure system <span>(langle Y rangle _P)</span> generated by the patch closure <i>Y</i> of <i>X</i> is the patch closure of <span>(langle X rangle _P)</span>. If <i>X</i> is contained in the set of nontrivial prime elements of <i>P</i> then <span>(langle X rangle _P)</span> is a frame and is a coherent algebraic frame if <i>X</i> is patch closed in <i>P</i>. Conversely, if the algebraic lattice <i>P</i> is coherent then its set of nontrivial prime elements is patch closed.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00815-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41700899","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":"Extremality, left-modularity and semidistributivity","authors":"Henri Mühle","doi":"10.1007/s00012-023-00814-8","DOIUrl":"10.1007/s00012-023-00814-8","url":null,"abstract":"<div><p>In this article we study the relations between three classes of lattices each extending the class of distributive lattices in a different way. In particular, we consider join-semidistributive, join-extremal and left-modular lattices, respectively. Our main motivation is a recent result by Thomas and Williams proving that every semidistributive, extremal lattice is left modular. We prove the converse of this on a slightly more general level. Our main result asserts that every join-semidistributive, left-modular lattice is join extremal. We also relate these properties to the topological notion of lexicographic shellability.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00814-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50488428","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":"Topological representation of double Boolean algebras","authors":"Prosenjit Howlader, Mohua Banerjee","doi":"10.1007/s00012-023-00811-x","DOIUrl":"10.1007/s00012-023-00811-x","url":null,"abstract":"<div><p>The article proves topological representations for some classes of double Boolean algebras (dBas). In particular, representation theorems characterising fully contextual and pure dBas are obtained. Duality results for fully contextual and pure dBas are also established.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44856585","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":"Quasi-Engel varieties of lattice-ordered groups","authors":"Michael R. Darnel","doi":"10.1007/s00012-022-00796-z","DOIUrl":"10.1007/s00012-022-00796-z","url":null,"abstract":"<div><p>We show that any ordered group satisfying the identity <span>([x_1^{k_1}, ldots , x_n^{k_n}] = e)</span> must be weakly abelian and that when <span>(x_i not = x_1)</span> for <span>(2 le i le n)</span>, <span>(ell )</span>-groups satisfying the identity <span>([x_1^n, ldots , x_k^n] = e)</span> also satisfy the identity <span>((x vee e)^{y^n} le (x vee e)^2)</span>. These results are used to study the structure of <span>(ell )</span>-groups satisfying identities of the form <span>([x_1^{k_1}, x_2^{k_2}, x_3^{k_3}] = e)</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47214233","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":"Congruence-simple multiplicatively idempotent semirings","authors":"Tomáš Kepka, Miroslav Korbelář, Günter Landsmann","doi":"10.1007/s00012-023-00807-7","DOIUrl":"10.1007/s00012-023-00807-7","url":null,"abstract":"<div><p>Let <i>S</i> be a multiplicatively idempotent congruence-simple semiring. We show that <span>(|S|=2)</span> if <i>S</i> has a multiplicatively absorbing element. We also prove that if <i>S</i> is finite then either <span>(|S|=2)</span> or <span>(Scong {{,textrm{End},}}(L))</span> or <span>(S^{op}cong {{,textrm{End},}}(L))</span> where <i>L</i> is the 2-element semilattice. It seems to be an open question, whether <i>S</i> can be infinite at all.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46213559","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":"Higher commutators in semigroups with zero","authors":"Jelena Radović, Nebojša Mudrinski","doi":"10.1007/s00012-023-00809-5","DOIUrl":"10.1007/s00012-023-00809-5","url":null,"abstract":"<div><p>We introduce the notion of the higher commutator of ideals in semigroups. For semigroups with zero, it is shown that the higher order commutator of Rees congruences is equal to the Rees congruence of the commutator of the corresponding ideals. We obtain that, for Rees congruences, higher order commutator is a composition of binary commutators. As a consequence, we prove that in semigroups with zero all four conditions of solvability, supernilpotency, nilpotency and nilpotency in the sense of semigroup theory, are equivalent.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00809-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49659077","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 representation of ordered semigroups by transformations of ordered sets","authors":"Michael Tsingelis","doi":"10.1007/s00012-023-00810-y","DOIUrl":"10.1007/s00012-023-00810-y","url":null,"abstract":"<div><p>A transformation of an ordered set <i>M</i> is an isotone mapping of <i>M</i> into <i>M</i>. By a representation of an ordered semigroup <i>S</i> by transformations of an ordered set <i>M</i> we mean a homomorphism of <i>S</i> into the set of transformations of <i>M</i>, i.e. (since the set of transformations of <i>M</i> is an ordered semigroup) an isotone mapping from <i>S</i> into the set of transformations of <i>M</i> preserving the operations. We prove that this type of representation leads to an “action” of <i>S</i> on <i>M</i> and so we introduce the notion of a left operand of <i>M</i> over <i>S</i>. Also we introduce the notions of a left operator pseudoorder on a left operand over <i>S</i> and a left operator homomorphism between left operands over <i>S</i>. We show that the concept of left operator pseudoorders on left operands over <i>S</i> plays an important role in the study of left operator homomorphisms of left operands over <i>S</i>. In the case of right operands over <i>S</i> dually definitions and results hold.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 2","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00810-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46952930","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}
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