Ordered vector spaces

Abstract

Simple consequences of these assumptions are: x>y implies x+z >y+z;x>y implies Xx>Xy for real positive scalarsX; x > 0 if and only if 0>-x. An important class of examples of such V's is due to R. Thrall ; we shall call these spaces lexicographic function spaces (LFS), defining them as follows: Let T be any simply ordered set ; let / be any real-valued function on T taking nonzero values on at most a well ordered subset of T. Let Vt be the linear space of all such functions, under the usual operations of pointwise addition and scalar multiplication, and define />0 to mean that/(/0) >0 if t0 is the first point of T at which/ does not vanish. Clearly Vt is an ordered vector space as defined above. What we shall show in the present note is that every V is isomorphic to a subspace of a Vt. 2. Dominance and equivalence. A trivial but suggestive special case of Vt is obtained when the set T is taken to be a single point. Then it is clear that Vt is order isomorphic to the real field. As will be shown later on, this example is characterized by the Archimedean property: if 0 x. Clearly the relation <C is nonreflexive, nonsymmetric, and transitive; and x<SCy implies x<y. For given x, yE V+, if neither of x and y dominates the other we say that x and y are equivalent, and write x~y. This relation is characterized by the existence of positive real X, ix such that Xx<y</*x, and it follows that it is indeed an equivalence relation on V+. We denote the class of elements of V+ equivalent to given x by [x].