I couldn't find any enough data about convex ideals while searching the internet .
Can any body help me?
Dear Enas,
A partially answer (working in a particular case) might be the following one. Assume that you have a partially ordered ring A, which is also a vector lattice over the real field (example: A=C(X), the space of all real continuous functions on a topological space X. The positive cone P is the cone of all nonnegative functions at each point of X). Define a positive convex ideal as being a convex ideal for which all its elements are positive. Then the sum of two positive convex ideals is a positive convex ideal. In fact, if 0
Is the sum of two convex ideals convex too? Yes it iis convex but it is to be changed as the linear sum of two convex ideals is a convex ideal. . . . Dr N V Nagendram
Dear Professor Octav Olteanu,
Thank you for your answer but will you provide some details about Riesz property?
I couldn't find enough useful details about it.
Dear Professor Enas Abueid,
The Riesz property is valid, in particular, in any vector lattice. Information on this subject was published in the classical monograph of H. H. Schaeffer: "Topological Vector Spaces", Springer-Verlag, Chapter 5. Another source is the monograph by Romulus Cristescu: 'Ordered Vector Spaces and Linear Operators", Academiei, Bucharest, and Abacus Press, Tunbridge Wells. For both these excellent books, there are several improved editions. More information is available on the Internet..
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