A homogenizable fishy poorset
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.
scientific article; zbMATH DE number 1271341