Minimal Lattice - Subspaces

In this paper the existence of minimal lattice-subspaces of a vector lattice E containing a subset B of E+ is studied (a lattice-subspace of E is a subspace of E which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology τ on E and E+ is τ -closed (especially if E is a Banach lattice with order continuous norm), then minimal lattice-subspaces with τ -closed positive cone exist (Theorem 2.5). In the sequel it is supposed that B = {x1, x2, . . . , xn} is a finite subset of C+(Ω), where Ω is a compact, Hausdorff topological space, the functions xi are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function β(t) = r(t) ‖r(t)‖1 where r(t) = ( x1(t), x2(t), . . . , xn(t) ) . If R(β) is the range of β and K the convex hull of the closure of R(β), it is proved: (i) There exists an m-dimensional minimal lattice-subspace containing B if and only if K is a polytope of Rn with m vertices (Theorem 3.20). (ii) The sublattice generated by B is an m-dimensional subspace if and only if the set R(β) contains exactly m points (Theorem 3.7). This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.