Extension theorems for linear lattices with positive algebraic basis

An ordered linear spaceV with positive wedgeK is said to satisfy extension property (E1) if for every subspaceL ₀ ofV such thatL ₀ ∩K is reproducing inL ₀, and every monotone linear functionalf ₀ defined onL ₀,f ₀ has a monotone linear extension to all ofV. A linear latticeX is said to satisfy extension property (E2) if for every sublatticeL ofX, and every linear functionalf defined onL which is a lattice homomorphism,f has an extensionf′ to all ofX which is also a linear functional and a lattice homomorphism. In this paper it is shown that a linear lattice with a positive algebraic basis has both extension property (E1) and (E2). In obtaining this result it is shown that the linear span of a lattice idealL and an extremal element not inL is again a lattice ideal. (HereX does not have to have a positive algebraic basis.) It is also shown that a linear lattice which possesses property (E2) must be linearly and lattice isomorphic to a functional lattice. An example is given of a function lattice which has property (E2) but does not have a positive algebraic basis. Yudin 12] has shown a reproducing cone in ann-dimensional linear lattice to be the intersection of exactlyn half-spaces. Here it is shown that the positive wedge in ann-dimensional archimedean ordered linear space satisfying the Riesz decomposition property must be the intersection ofn half-spaces, and hence the space must be a linear lattice with a positive algebraic basis. The proof differs from those given for the linear lattice case in that it uses no special techniques, only well known results from the theory of ordered linear space.