An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL.
A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism.
These properties were introduced by Schmidt in .
In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic
to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM.
An analogous theorem is derived for Riesz spaces with extension property (E2).
We sought to compare the efficacy of the stationary Markov model and conventional ordination techniques in describing compositional and structural changes in forest communities along natural and manmade spatial gradients at two scales, local and regional. Vegetation abundance and structure data are from six sites spanning a spatial gradient in the Great Lakes-St. Lawrence forests near Sudbury, Ontario, Canada. Ordination did not detect slope-related local gradients despite the general trend that, as distance from the pollution source increases, vegetation along the slopes begins to display Markovian spatial dynamics. We suggest that this is due to information loss resulting from static ordination analyses: information regarding transitions between observations along the natural ordering of quadrats is not maintained. Both ordination techniques and the Markov analyses detected strong regional pollution-induced gradients in abundance and structure.