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Relations related to betweenness: Their structure and automorphisms. (English) Zbl 0896.08001
Mem. Am. Math. Soc. 623, 125 p. (1998).
This monograph was inspired by the study of Jordan groups in the theory of permutation groups. The authors set out to systematise the underlying structures, such as the tree-like structures used by P. J. Cameron [J. Lond. Math. Soc., II. Ser. 27, 238-247 (1983; Zbl 0519.20004)], but the relations they constructed turned out to be of intrinsic interest, and to be the underlying structures of other mathematical objects, such as $$\Lambda$$-trees and certain general topological spaces. The basic idea is that of a semilinear order (or, more precisely, an upper semilinear order). This is a partial ordering, which in addition to the usual requirements of antisymmetry and transitivity, has the properties that the principal filter generated by any element is a chain, and that any two elements have a common upper bound. The semilinear orders studied in this work are also required to be nonlinear and to have no minimal and maximal elements. The importance of such orders in group theory is that they may have relatively 2-transitive automorphism groups. Associated with these semilinear orderings we have two ternary relations and a quaternary relation. The properties of the first, called abstract chain sets, or $$C$$-sets are derived from those of chains in a semilinear ordering. These are axiomatised by a ternary relation $$C(\alpha;\beta,\gamma)$$ satisfying the following conditions: (C1) $$C(\alpha; \beta,\gamma)\to C(\alpha;\gamma,\beta)$$; (C2) $$C(\alpha; \beta,\gamma)\to \neg C(\beta;\alpha,\gamma)$$; (C3) $$C(\alpha; \beta,\gamma)\to C(\alpha;\delta,\gamma)\lor C(\delta;\beta,\gamma)$$; (C4) $$\alpha\neq \beta\to C(\alpha; \beta,\beta)$$. The second ternary relation, called a $$B$$-relation, is derived from betweenness in a semilinear linear order. Its axioms are: (B1) $$B(\alpha; \beta,\gamma)\to B(\alpha;\gamma,\beta)$$; (B2) $$B(\alpha; \beta,\gamma)\land B(\beta;\alpha,\gamma)\to \alpha=\beta$$; (B3) $$B(\alpha; \beta,\gamma)\to B(\alpha;\beta,\delta)\lor B(\alpha;\gamma,\delta)$$. $$B$$-sets have a tree-like structure, and so have directions. From these is derived the notion of an abstract set of directions, or $$D$$-set. The quaternary relation $$D(\alpha,\beta;\gamma,\delta)$$ axiomatising these satisfies: (D1) $$D(\alpha,\beta;\gamma,\delta)\to D(\beta,\alpha;\gamma,\delta)\land D(\alpha,\beta;\delta,\gamma)\land D(\gamma,\delta;\alpha,\beta)$$. (D2) $$D(\alpha,\beta;\gamma,\delta)\to \neg D(\alpha,\gamma;\beta,\delta)$$; (D3) $$D(\alpha,\beta;\gamma,\delta)\to D(\epsilon,\beta;\gamma,\delta)\lor D(\alpha,\beta;\gamma,\epsilon)$$; (D4) $$(\alpha\neq\gamma\land \beta\neq\gamma)\to D(\alpha,\beta;\gamma,\gamma)$$. Parts II-V of the monograph are devoted to deriving properties of these relations and giving concrete examples. The last part gives applications, with particular emphasis on the theory of highly transitive groups, but not forgetting other topics such as topology.
[Reviewers remark: An index or, at the very least, page numbers for the sections, would have made life much easier for those of us whose short-term memories are not what they used to be!].

##### MSC:
 08A02 Relational systems, laws of composition 06A06 Partial orders, general 03C35 Categoricity and completeness of theories 20E08 Groups acting on trees 54F50 Topological spaces of dimension $$\leq 1$$; curves, dendrites 08-02 Research exposition (monographs, survey articles) pertaining to general algebraic systems 20B07 General theory for infinite permutation groups
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