Relations related to betweenness

Abstract

In the last chapter we defined linear betweenness relations, circular (or cyclic) orders and separation relations from a linear order and studied their groups of automorphisms. The automorphism group of a linear order has already been studied in detail in Chapter 9. That of a circular order can be understood best in terms of the linear order obtained by deleting a point. The groups of automorphisms of a linear betweenness relation and of a separation relation are simply the groups of order-preserving or order-reversing transformations on a linearly ordered and cyclically ordered set respectively. In this chapter we introduce four more relations related to betweenness which will lead us to the classification of primitive Jordan groups that have primitive Jordan sets. Everything discussed in this chapter has been discussed in greater detail and rigour in Adeleke & Neumann (1996c). The arguments used in this chapter are very geometric and we encourage the reader to draw pictures. Note however that our semilinear orders grow upwards rather than downwards, contrary to the convention followed in Adeleke & Neumann (1996c).