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A note on barreledness in locally convex cones

Locally convex cones are generalization of locally convex spaces. The assertion, whether a barreled cone is an upper-barreled cone or not, was posed as a question in [A. Ranjbari, H. Saiflu, Projective and inductive limits in locally convex cones, J. Math. Anal. Appl. 332 (2) (2007) 1097-1108]. In this paper, we show that a barreled locally convex cone is not necessarily upper-barreled.