Cones and orderings; order convexity; order units; approximate order units; bases. Positive linear mappings and functionals; extension and separation theorems; decomposition of linear functionals into positive linear functionals. Vector lattices; basic theory. Norms and orderings; duality of ordered spaces; (approximate) order unit spaces; base normed spaces. Normed and Banach lattices; AM-spaces, AL-spaces; Kakutani theorems for AM-spaces and ALspaces. Matrix ordered spaces: matricially normed spaces; matricial ordered normed spaces; matrix order unit spaces; Arveson-Hahn-Banach extension theorem.
1.W.A. J. Luxemburg and A.C. Zaanen, \Riesz Spaces", Elsevier, 1971.
2. A.C. Zaanen, \Introduction to operator theory in Riesz spces (Vol 1 & Vol 2)", Springer, 1997