SOME MONOTONICITY PROPERTIES IN CERTAIN s-NORMED (0 < s < 1) AND F-NORMED LATTICES

In this paper, we study two variants of strict monotonicity in nonBanach F-lattices with an emphasis on function spaces, in particular, for (nonBanach) Orlicz spaces. First it is shown that in Dedekind complete F-normed lattices strict monotonicity coincides with orthogonal strict monotonicity. Criteria for strict monotonicity for s-concavifications (0 < s < 1) of normed Kothe spaces as well as for Orlicz s-normed spaces generated by s-convex Orlicz functions and for F-normed Orlicz spaces generated by monotone Orlicz functions and equipped with the Mazur-Orlicz F-norm are given. In the case of the s concavification of normed lithe spaces also criteria for orthogonal lower and upper local uniform monotonicities as well as for orthogonal uniform monotonicity are given. Some examples are also presented.