  In Orlicz spaces generated by convex Orlicz functions a family of norms
generated by some lattice norms in $\mathbb{R}^{2}$ are defined and studied.
This family of norms includes the family of the p-Amemiya norms ($1\leq
p\leq\infty$) studied in [10-11], [14-15] and [20]. Criteria for strict
monotonicity, lower and upper local uniform monotonicities and uniform
monotonicities of Orlicz spaces and their subspaces of order continuous
elements, equipped with these norms, are given in terms of the generating
Orlicz functions, and the lattice norm in $\mathbb{R}^{2}$. The problems of
strict convexity and of the existence of order almost isometric as well as of
order isometric copies in these spaces are also discussed.
