If E and F are locally convex lattices it is shown that the closure of the projective cone equals the biprojective cone for the projective topology on E 0 F and that this result cannot be extended to arbitrarily ordered locally convex spaces. The question of when these cones are b-cones is considered and applications are given to the tensor product of unconditional Schauder bases. In this note we investigate some properties of the projective and biprojec- tive cones in the tensor product of two locally convex lattices. In section 1 it is shown that the closure of the projective cone for the projective (n7) topology is the biprojective cone and an example is given to show that this result does not extend to arbitrary ordered locally convex spaces. The next section is concerned with the question of when the projective cone is a b-cone in the tensor product of ordered Banach spaces and we show that it is a b-cone for the projective topology but it is not, in general, a b-cone for the topology of bi-equicontinuous convergence. In the final section of the paper we use an order theoretic approach to study the tensor product of unconditional bases in Banach spaces. We show that the cone generated by the tensor product basis is just the biprojective cone and we give order criteria on when this basis is unconditional. Using these notions and the results of sections 1 and 2, we give examples of when the tensor product basis is unconditional and when it is conditional. Notation and terminology for this paper follows (12) and (15). 1. Cones in the Tensor Product. Let E and F be locally convex spaces