Local condition as a crucial tool for local structure of Calderón–Lozanovskiĭ spaces

We shall give a full criterion for a point x∈Eφ to be a point of order continuity of Calderón–Lozanovskiĭ spaces Eφ. In order to get that result we introduce the new local condition which appears to be necessary and sufficient for this goal. Next we study the structure of points of lower local uniform monotonicity (LLUM for short). We give a result concerning general Köthe sequence spaces and we present a characterization for LLUM points in Calderón–Lozanovskiĭ spaces.