  We describe pure braided versions of Thompson's group F. These groups, $BF$
and $\hat{BF}$, are subgroups of the braided versions of Thompson's group V,
introduced by Brin and Dehornoy. Unlike V, elements of F are order-preserving
self-maps of the interval and we use pure braids together with elements of F
thus preserving order. We define these groups and give normal forms for
elements and describe infinite and finite presentations of these groups.
