Order bornological spaces and order ultrabornological spaces

Ordered locally convex spaces is an important classes of spaces in the theory of ordered topological vector spaces just as locally convex spaces in the theory of topological vector spaces. Some special classes of ordered locally convex spaces such as order infrabarrelled spaces have been studied. The purpose of this paper is to initiate the study of order bornological spaces and their fundamental properties. In addition, order ultrabornological spaces is also investigated as an important special class of order bornological spaces.