Given a set L subset of R-X of functions defined on X, we consider abstract monotone (or, for short, L-monotone) multivalued operators T : X paired right arrows L. We extend the definition of enlargement of monotone operators to this framework and study semicontinuity properties of these mappings. We prove that sequential outer semicontinuity, which holds for maximal monotone operators and their enlargements in the classical case (i.e., when L = X* and X is a Banach space), holds also in our abstract setting. We also show through examples that some properties, known to hold in the classical case, may no longer be valid in the abstract setting. One of these properties is the maximality of the subdifferential and another one is the lack of inner semicontinuity of (point-to-set) monotone operators in the interior of their domain. We also focus on the structure of both the abstract subdifferential and the abstract epsilon-subdifferential. This is a key question in abstract convexity because these sets may be very large for certain choices of L and therefore it is important to be able to represent them by means of some special elements of the set of "affine" functions induced by L.
Given a generic dual program we discuss the absence of duality gap for a family of Lagrange-type functions. We obtain necessary conditions that become sufficient ones under some additional assumptions. We also give examples of Lagrangetype functions for which this sufficient conditions hold.