A homogenizable fishy poorset
type: Item
instance of: scholarly article
title: A homogenizable fishy poorset
zbMATH Open document ID: 0928.03046
author: Jacinta Covington
author: D. A. Thomas
published in: Journal of Algebra
publication date: 2000-01-09T00:00:00Z
review text: The authors build a partially ordered set \((\Omega,<)\) which is a poorset (it contains no diamonds, that is, no two incomparable elements can have both a common upper bound and a common lower bound) and is fishy. The latter means that any two incomparable elements have either a common upper bound or a common lower bound, and that if \(a,b\) have no lower bound and \(a,c\) have no upper bound, then \(b<c\). In addition, they ensure that \((\Omega,<)\) has a strong homogeneity condition on triples: that is, the automorphism group \(G\) is transitive on (the set of) triples of the form \((a,b,c)\) where \(a<b<c\), and on those where \(b<a\) and \(a\wedge c\), and on those where \(a<b\) and \(a\vee c\). (Here \(a\wedge c\) means that \(a,c\) are incomparable and bounded above, and \(a\vee c\) that they are incomparable and bounded below). The existence of such partial orders arose form work by P. M. Neumann on a graph on the suborbits of a primitive permutation group (see Question 3 of \textit{P. M. Neumann} [``Coprime suborbits of transitive permutation groups'', J. Lond. Math. Soc. (2) 47, No.~2, 285-293 (1993; Zbl 0815.20003)]). The construction technique is model-theoretic. The authors use a variant of Fraïssé's Amalgamation Theorem, due to \textit{J. Covington} [``Homogenizable relational structures'', Ill. J. Math. 34, No. 4, 731-743 (1990; Zbl 0706.03030)]. They describe a class of finite fishy structures in a language with binary relation symbols \(<,\wedge,\vee\), show that this class only fails to have the amalgamation property due to finitely many `bad' pictures, and deduce that it is possible to add finitely many quaternary relation symbols to obtain an amalgamation class. Amalgamation now gives a countable homogeneous structure whose reduct to the language with just the relation symbol \(<\) is the required fishy poorset.
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 20B07
Mathematics Subject Classification ID: 03C52
zbMATH DE Number: 1271341
zbMATH Keywords: homogeneous structure
zbMATH Keywords: fishy poorset
zbMATH Keywords: amalgamation property
zbMATH Keywords: amalgamation class
reviewed by: H. D. Macpherson
MaRDI profile type: Publication
full work available at URL: jabr.1997.7187
OpenAlex ID: W2068578639
cites work: Homogenizable relational structures
cites work: Coprime Suborbits of Transitive Permutation Groups
cites work: Sur l'extension aux relations de quelques propriétés des ordres
DOI: 10.1006/JABR.1997.7187
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description: scientific article; zbMATH DE number 1271341