Order isomophisms between Riesz spaces

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms [Formula: see text] where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism [Formula: see text] . We prove the existence of a homeomorphism [Formula: see text] that maps the graph of any [Formula: see text] onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms [Formula: see text] . (Here [Formula: see text] is the space of all continuous functions [Formula: see text] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms [Formula: see text] of E and F onto order dense Riesz subspaces of [Formula: see text] and an order isomorphism [Formula: see text] such that [Formula: see text] ([Formula: see text] ).