  In this paper we begin an exploration of several domination-related
parameters (among which are the total, restrained, total restrained, paired,
outer connected and
  total outer connected domination numbers) in the generalized lexicographic
product (GLP for short) of graphs. We prove that for each GLP of graphs there
exist several equality chains containing these parameters. Some known results
on standard lexicographic product of two graphs are generalized or/and
extended. We also obtain results on well $\mu$-dominated GLP of graphs, where
$\mu$ stands for any of the above mentioned domination parameters. In
particular, we present a characterization of well $\mu$-dominated GLP of graphs
in the cases when $\mu$ is the domination number or the total domination
number.
