Lyaletskyy O. Continuity of a function in intensional models of lambda-like calculi

The dissertation is devoted to investigations of some special notions of continuity of a function in the view of constructing, on their base, new lambda-calculus models. More precisely: In the paper, 3 new natural notions of convergence of a direction on a partially ordered set with "enough" number of supremums and infimums are introduced. On the base of these 3 new notions, 3 new similar notions of continuity of a function are introduced as the property to preserve the corresponding limits of the directions. There was made the comparative analysis of these 3 notions with their well-known analogues, such as Scott continuity, (o)-continuity, and topological continuity. Abstract order-theoretical characterization of these 3 notions is constructed. By means of these characterization theorems, the new notions of continuity of a function were tested on possibility of the construction of new lambda-models in accordance with the Scott-Koymans method. It appeared that one of these notions as well as the notion of an (o)-continuity of a function induct no lambda-models except the trivial one, but 2 other notions lead to the constriction of new "continuous" lambda-models. An example of "continuous" but non-topologizable lambda-model was constructed.