that a lower semicontinuous set-valued operator satisfying a generalized monotonicity property locally is actually globally generalized monotone. We are concerned on strictly quasimonotone, pseudomonotone as well as strictly pseudomonotone set-valued operators, and these results extend and improve the results obtained in the single-valued case in [@La] and [@KaPiLa].

We study generalized monotonicity properties on certain dense subsets that we call self segment-dense. By an example we show that this new concept differs from that of a segment-dense set introduced by Dinh The Luc [@DTL] in the context of densely quasimonotone, respectively densely pseudomonotone operators.

Two counterexamples circumscribe the area of our research. We show that the local (generalized) monotonicity of an upper semicontinuous set-valued operator does not imply its global counterpart (Example \[ex2\]). Then we also provide an example of lower semicontinuous set-valued operator, which is locally quasimonotone on its domain but is not globally quasimonotone (Example \[ex1\]). Hence, we deal with locally strictly quasimonotone, locally pseudomonotone, respectively locally strictly pseudomonotone lower semicontinuous set-valued operators.

The outline of the paper is the following. In the next section we introduce the necessary apparatus that we need to obtain our results. We also introduce the notion of a self segment-dense set and 