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UDC 515.12 We show that a continuous open linear operator preserves the completeness and barreledness in locally convex cones. Specially, we prove some relations between an open linear operator and its adjoint in uc-cones (locally convex cones which their convex quasi-uniform structures are generated by one element).

In this paper, we verify a relation between the symmetric continuity of lower semicontinuous extended seminorms on a locally convex cone with respect to the symmetric topology and barreledness of the locally convex cone. The assertion, whether a barreled cone is upper-barreled or not, was posed as a question in Ranjbari and Saiflu (J Math Anal Appl 332(2):1097–1108, 2007). We show that a barreled locally convex cone is not necessarily upper-barreled.

In this paper we define bornological and b-bornological convergence in locally convex cones. We give some characterizations for these convergences. In special case of locally convex topological vector spaces, both these concepts reduce to the known concept of Mackey-convergence. We investigate the relationship between bornological cones and bornological convergence. Finally, we investigate bornological completeness.