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Partial orders of the group of automorphisms of the real line. (English) Zbl 0766.06015
Algebra, Proc. Int. Conf. Memory A. I. Mal’cev, Novosibirsk/USSR 1989, Contemp. Math. 131, Pt. 1, 197-207 (1992).
[For the entire collection see Zbl 0745.00032.]
Let $$A(R)$$ be the group of all order-preserving permutations of the real line $$R$$. Under the pointwise order, $$A(R)$$ is a lattice-ordered group. Let us denote by $$S$$ the system of all non-trivial partial orders on $$A(R)$$ under which $$A(R)$$ is a partially ordered group. The system $$S$$ is partially ordered in a natural way.
The results of this very interesting paper are as follows. (a) $$S$$ contains exactly two elements under which $$A(R)$$ is lattice-ordered, namely the pointwise order and its inverse. (b) $$S$$ has exactly 16 minimal elements. (c) $$S$$ has exactly 40 maximal elements.

##### MSC:
 06F15 Ordered groups