Order Schauder bases in Banach lattices

We introduce and study the notion of an order Schauder basis of a vector lattice E by replacing the norm convergence in the definition of a Schauder basis with the order convergence in E. By a bibasis of a Banach lattice E we mean a sequence which is both a Schauder basis and an order Schauder basis of E  . We find necessary and sufficient conditions for a system to be a bibasis, and extend some known theorems on Schauder bases to the setting of bibases. Then we show that the Haar system is a bibasis of LpLp with 1