OrderPub Date : 2024-07-13DOI: 10.1007/s11083-024-09676-8
Maria Manuel Clementino, George Janelidze
{"title":"Effective Descent Morphisms of Filtered Preorders","authors":"Maria Manuel Clementino, George Janelidze","doi":"10.1007/s11083-024-09676-8","DOIUrl":"https://doi.org/10.1007/s11083-024-09676-8","url":null,"abstract":"<p>We characterize effective descent morphisms of what we call filtered preorders, and apply these results to slightly improve a known result, due to the first author and F. Lucatelli Nunes, on the effective descent morphisms in lax comma categories of preorders. A filtered preorder, over a fixed preorder <i>X</i>, is defined as a preorder <i>A</i> equipped with a profunctor <span>(Xrightarrow A)</span> and, equivalently, as a set <i>A</i> equipped with a family <span>((A_x)_{xin X})</span> of upclosed subsets of <i>A</i> with <span>(x'leqslant xRightarrow A_xsubseteq A_{x'})</span>.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-07-09DOI: 10.1007/s11083-024-09667-9
Jingjing Ma
{"title":"Partially Ordered Fields and Integral Domains","authors":"Jingjing Ma","doi":"10.1007/s11083-024-09667-9","DOIUrl":"https://doi.org/10.1007/s11083-024-09667-9","url":null,"abstract":"<p>The article studies the division closed partial orders on fields that are algebraic over the field of rational numbers. In particular, the maximal partial orders are described using embeddings from the given field to the field of complex numbers. The <span>(O^{*})</span>-fields that are not finite dimensional over <span>(mathbb {Q})</span> are studied in Section 2 and the <span>(n^{th})</span>-root function over totally ordered fields is considered in Section 3.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141567922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-07-04DOI: 10.1007/s11083-024-09674-w
Dömötör Pálvölgyi, Balázs Patkós
{"title":"Projective and External Saturation Problem for Posets","authors":"Dömötör Pálvölgyi, Balázs Patkós","doi":"10.1007/s11083-024-09674-w","DOIUrl":"https://doi.org/10.1007/s11083-024-09674-w","url":null,"abstract":"<p>We introduce two variants of the poset saturation problem. For a poset <i>P</i> and the Boolean lattice <span>(mathcal {B}_n)</span>, a family <span>(mathcal {F})</span> of finite subsets of <span>(mathbb {N})</span>, not necessarily from <span>(mathcal {B}_n)</span>, is <i>projective</i> <i>P</i><i>-saturated</i> if (i) it does not contain any strong copies of <i>P</i>, (ii) for any <span>(Gin mathcal {B}_nsetminus mathcal {F})</span>, the family <span>(mathcal {F}cup {G})</span> contains a strong copy of <i>P</i>, and (iii) for any two different <span>(F,F'in mathcal {F})</span> we have <span>(Fcap [n]ne F'cap [n])</span>. Ordinary strongly <i>P</i>-saturated families, i.e., families <span>(mathcal {F}subseteq mathcal {B}_n)</span> satisfying (i) and (ii), automatically satisfy (iii) as they lie within <span>(mathcal {B}_n)</span>. We study what phenomena are valid both for the ordinary saturation number <span>(text {sat}^{*}(n,P))</span> and the projective saturation number <span>(top hspace{-10pt}top text {sat}(n,P))</span>, the size of the smallest projective <i>P</i>-saturated family. Note that the projective saturation number might differ for a poset and its dual. Motivated by this, we introduce an even more relaxed and symmetric version of poset saturation, <i>external saturation</i>. We conjecture that all finite posets have bounded external saturation number, and prove this in some special cases.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-06-27DOI: 10.1007/s11083-024-09675-9
Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz
{"title":"Sizes of Flat Maximal Antichains of Subsets","authors":"Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz","doi":"10.1007/s11083-024-09675-9","DOIUrl":"https://doi.org/10.1007/s11083-024-09675-9","url":null,"abstract":"<p>This is the second of two papers investigating for which positive integers <i>m</i> there exists a maximal antichain of size <i>m</i> in the Boolean lattice <span>(B_n)</span> (the power set of <span>([n]:={1,2,dots ,n})</span>, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only <i>l</i>-sets and <span>((l+1))</span>-sets for some <i>l</i>.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-06-19DOI: 10.1007/s11083-024-09672-y
Tatsuya Goto
{"title":"The Comparability Numbers and the Incomparability Numbers","authors":"Tatsuya Goto","doi":"10.1007/s11083-024-09672-y","DOIUrl":"https://doi.org/10.1007/s11083-024-09672-y","url":null,"abstract":"<p>We introduce new cardinal invariants of a poset, called the comparability number and the incomparability number. We determine their value for well-known posets, such as <span>(omega ^omega )</span>, <span>(mathcal {P}(omega )/textrm{fin})</span>, the Turing degrees <span>(mathcal {D})</span>, the quotient algebra <span>(textsf {Borel}(2^omega )/textsf {null})</span>, the ideals <span>(textsf {meager})</span> and <span>(textsf {null})</span>. Moreover, we consider these invariants for the Rudin-Keisler ordering of the nonprincipal ultrafilters on <span>(omega )</span>. We also consider these invariants for ideals on <span>(omega )</span> and on <span>(omega _1)</span>.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141506517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-05-27DOI: 10.1007/s11083-024-09671-z
Adam Přenosil
{"title":"Pointed Lattice Subreducts of Varieties of Residuated Lattices","authors":"Adam Přenosil","doi":"10.1007/s11083-024-09671-z","DOIUrl":"https://doi.org/10.1007/s11083-024-09671-z","url":null,"abstract":"<p>We study the pointed lattice subreducts of varieties of residuated lattices (RLs) and commutative residuated lattices (CRLs), i.e. lattice subreducts expanded by the constant <span>(textsf{1})</span> denoting the multiplicative unit. Given any positive universal class of pointed lattices <span>(textsf{K})</span> satisfying a certain equation, we describe the pointed lattice subreducts of semi-<span>(textsf{K})</span> and of pre-<span>(textsf{K})</span> RLs and CRLs. The quasivariety of semi-prime-pointed lattices generated by pointed lattices with a join prime constant <span>(textsf{1})</span> plays an important role here. In particular, the pointed lattice reducts of integral (semiconic) RLs and CRLs are precisely the integral (semiconic) semi-prime-pointed lattices. We also describe the pointed lattice subreducts of integral cancellative CRLs, proving in particular that every lattice is a subreduct of some integral cancellative CRL. This resolves an open problem about cancellative CRLs.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168072","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-05-07DOI: 10.1007/s11083-024-09669-7
Geir Agnarsson, Neil Epstein
{"title":"On Posets, Monomial Ideals, Gorenstein Ideals and their Combinatorics","authors":"Geir Agnarsson, Neil Epstein","doi":"10.1007/s11083-024-09669-7","DOIUrl":"https://doi.org/10.1007/s11083-024-09669-7","url":null,"abstract":"<p>In this article we first compare the set of elements in the socle of an ideal of a polynomial algebra <span>(K[x_1,ldots ,x_d])</span> over a field <i>K</i> that are not in the ideal itself with Macaulay’s inverse systems of such polynomial algebras in a purely combinatorial way for monomial ideals, and then develop some closure operational properties for the related poset <span>({{mathbb {N}}_0^d})</span>. We then derive some algebraic propositions of <span>(Gamma )</span>-graded rings (a natural generalization of the usual <span>({mathbb {Z}})</span>-grading where <span>(Gamma )</span> is a monoid) that then have some combinatorial consequences. Interestingly, some of the results from this part that uniformly hold for polynomial rings are always false when the ring is local. We finally delve into some direct computations, in relation to a given term order of the monomials, for general zero-dimensional Gorenstein ideals, and we deduce a few explicit observations and results for the inverse systems from some recent results about socles.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140888192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-04-06DOI: 10.1007/s11083-024-09668-8
{"title":"On the Width of the Cartesian Product of Ordinals","authors":"","doi":"10.1007/s11083-024-09668-8","DOIUrl":"https://doi.org/10.1007/s11083-024-09668-8","url":null,"abstract":"<h3>Abstract</h3> <p>The ordinal invariants, i.e., maximal order type, height, and width, are measures of a well quasi-ordering (wqo) based on the ordinal rank of the trees of its bad sequences, strictly decreasing sequences, and antichain sequences, respectively. Complex wqos are often built from simpler wqos through basic constructions such as disjoint sum, direct sum, cartesian product, and higher-order constructions like powerset or sequences. One main challenge is to compute the ordinal invariants of such wqos compositionally. This article focuses on the width of the cartesian product of wqos, for which no general formula is known. The particular case of the cartesian product of two ordinals has already been solved by Abraham in 1987, using the methods of residuals. We introduce a new method to get lower bounds on width, and apply it to the width of the cartesian product of finitely many ordinals, thus generalizing Abraham’s result. Finally, we leverage this result to compute the width of a generic family of elementary wqos that is closed under cartesian product.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-04-03DOI: 10.1007/s11083-024-09664-y
{"title":"Well-quasi-ordering and Embeddability of Relational Structures","authors":"","doi":"10.1007/s11083-024-09664-y","DOIUrl":"https://doi.org/10.1007/s11083-024-09664-y","url":null,"abstract":"<h3>Abstract</h3> <p>At the end of the forties, Fraïssé, following Cantor, Hausdorff and Sierpinski, highlighted the role of the embeddability quasi-order in the theory of relations. Since then, many results illustrating this role have been obtained (a large account was included in Fraïssé’s book <em>Theory of Relations</em>). In this paper, I present a selection of results centered on the notion of well-quasi-order (wqo). I mention several problems on wqo and hereditary classes of relational structures; some of these problems go back to the seventies.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140560378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
OrderPub Date : 2024-03-14DOI: 10.1007/s11083-024-09665-x
Wen Ma, Jinquan Luo
{"title":"Constructions of Codes with Weighted Poset Block Metrics","authors":"Wen Ma, Jinquan Luo","doi":"10.1007/s11083-024-09665-x","DOIUrl":"https://doi.org/10.1007/s11083-024-09665-x","url":null,"abstract":"<p>Weighted poset block metric is a generalization of two types of metrics: one is weighted poset metric introduced by Panek and Pinheiro (2010) and the other is metric for linear error-block codes introduced by Feng and Hickernell (2006). This type of metrics includes many classical metrics such as Hamming metric, Lee metric, poset metric, pomset metric, poset block metric, pomset block metric and so on. In this work, we focus on constructing new codes under weighted poset block metric from given ones. Some basic properties such as minimum distance and covering radius are studied.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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