Lushness, Numerical Index 1 and the Daugavet Property in Rearrangement Invariant Spaces

We show that for spaces with 1–unconditional bases lushness, the alternative Daugavet property and numerical index 1 are equivalent. In the class of rearrangement invariant (r.i.) sequence spaces the only examples of spaces with these properties are c0, ℓ1andℓ∞. The only lush r.i. separable function space on [0, 1] is L1[0, 1]; the same space is the only r.i. separable function space on [0, 1] with the Daugavet property over the reals.

The Determination on Minimal Covering of Regular Separable Function Sets in Partial Four-Valued Logic

In the structure theory of many-valued logic function, the decision and constitution of the Sheffer function is a very important problem, which is reduced to the decision of the minimal covering of precomplete sets in the many-valued logic function sets. According to the completeness theory in partial k-valued logic and the similar relationship theory among precomplete sets, in this paper, the methods of determination on the minimal covering of regular separable function sets are found out, and the minimal covering of regular separable function sets in partial four-valued logic are decided.