Let X be a linear space over K, K=R or K=C and let for n>1 \rho_i be s-convex semimodular defined on X for any i\in{ 1,...,n-1}. Put \rho=\max_{ 1\leq i \leq n-1}\{ \rho_i\} and X_{ \rho}= { x \in X: \rho(dx) < \infty for some d > 0 }. In this paper we define a new class of s-norms (norms if s=1) on X_{ \rho}. In particular, our defintion generalizes in a natural way the Orlicz-Amemiya and Luxemburg norms defined for s-convex semimodulars. Then, we investigate order continuous, the Fatou Property and various monotonicity properties of semimodular spaces equipped with these s-norms.
Categories: math.FA