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Oriented and nonoriented linear orders. (English) Zbl 0358.06001
Let \(S\neq \emptyset\). A ternary relation \(B\subset S^3\) is said to be a betweenness relation in \(S\) whenever the following five axioms are satisfied:
\[ B(abc)\lor B(bca)\lor B(cab), \quad B(aba)\Rightarrow a=b,\quad B(abc)\Rightarrow B(cba), \]
\[ B(abc) \land B(acd)\Rightarrow B(bcd),\quad B(abc) \land B(bcd) \land b\neq c\Rightarrow B(acd). \]
The pair \(\{\leq,\geq\}\) of mutually inverse linear orders in \(S\) is referred to as a non-oriented linear order. The paper concerns the relationship between non-oriented linear orders on the one hand, and betweenness relations on the other. The author proves that every non-oriented linear order \(\{\leq,\geq\}\) in \(S\) generates a betweenness relation \(B\) which is definable by means of \(\{\leq,\geq\}\), and, conversely, every betweenness relation \(B\) in \(S\) together with a fixed triplet of points \(a,b,c \in S\) generates a non-oriented linear order \(\{\leq,\geq\}\) which is definable in terms of \(B, a, b, c\). Moreover, there are two mutually inverse definitional functions \(\Phi\) and \(\Psi\), from the class of all the betweenness relations in \(S\) to the class of all the non-oriented linear orders in \(S\), and conversely. Each of the above axioms is proved to be independent of the remaining four. For related results see [Z. Piesyk, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 25, 667–670 (1977; Zbl 0371.06001)].

06A05 Total orders
08A02 Relational systems, laws of composition