Monotonicity Properties and Order Smoothness in Banach Lattices

Some versions of local uniform monotonicity and order uniform smoothness of Banach lattices are studied. We show an easy method of constructing lattice norms in K & ouml;the sequence spaces which are lower but not upper locally uniformly monotone. We also give an example of a K & ouml;the sequence space which is upper but not lower locally uniformly monotone. This shows that the lower and upper versions of local uniform monotonicity are different from each other. For local order uniform smoothness we establish a counterpart of the well-known & Scaron;mulian test on differentiability of norms and prove results on duality between local order uniform smoothness and local uniform monotonicity. We also give a proof of the equality of modulus of monotonicity of a lattice and its second dual.