## 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)].

### MSC:

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

Zbl 0371.06001